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NOTES ON 
HYDRAULIC MEASUREMENTS, 

PREPARED FOR THE USE 0? 

STUDENTS IN CIVIL AND SANITARY ENGINEERING 

AT THE 

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 

BY 

DWIGHT PORTER . 

1909 . 


4 


Copyright, 1909, by Dwight Porter 













251011 


a 


TABLE OP CONTENTS ■ 

Art . 

1 . Introduction . 

2. Relation between Discharge, Mean Velocity and Area. 

3 . Choice of Location for Discharge Measurements . 

4 . Gage . 

5. Soundings . 

6, Interpolation of Soundings . 

7 a Computation of Area, from Soundings in a Single Cross-Section . 

8. Computation of Mean Area from Soundings in Several Cross-Sections. 

9 » Complex Motion of flowing Water * 

10. Velocity Pulsations. 

11. General Law of Variation of Velocities past a Cross-Section. 

12. Velocity Measurements at Numerous Points Uniformly Distributed 
over the Cross-Section « 

13. Reduction Coefficients., 

14. Velocity Measurements at but One Point in the Cross-Section . 

15. Velocity Measurements in Each of a Series of Verticals. 

16. Vertical Velocity Curves • 

17. Determination of Reduction Coefficients for Vertical Velocity Curves. 

18. Choice of Observation Points in Verticals. 

19 o Velocity Measurements at or near Surface . 

20 • Velocity Measurements at Probable Depth of Mean Velocity past 

Vertical. 

21. Velocity Measurements at Mid-Depth in Vertical. 

22. Single-Point Velocity Measurements at Sundry Depths in Verticals. 







b 


23. Velocity Measurements at Two Points in Vertical. 

24. Velocity Measurements at Three Points in Vertical. 

25. Determination of Mean Velocity past Vertical from Curve. 

26. Integration of Velocities past Vertical. 

27. Method of Measurement used by Kydrcgraphic Department of Hungary . 

28. Measurement of Mean Velocity past Vertical by Rod or Tube Fleets. 

29. Proper Spacing of Verticals. 

30 * "Flanking’ 5 Method of Measuring Velocities . 

31. Continuous Integration over Entire Cross-Section. 

32. Transverse Velocity Curve. 

33. • Conception of Discharge, as Represented by a Solid. 

34. Subdivision of Solid into Slices. 

35. KarX&eher’e Graphical Method of Computing Discharge. 

36. Procedure in Computing Discharge for Irregularly Distributed 
Verticals, as in Float Measurements. 

37 0 Procedure *in Computing Discharge for Uniformly Distributed Verticals. 

38. Special Methods of Computation Allowable for Rectangular Cross- 
Sections . 

39. Discharge or "Station Rating" Curves. 

40. Discharge Tables. 

41. Extension of Discharge Curve beyond Limits of Observations. 

42. Area, Mean Velocity and Discharge Curves: Theoretical Considerations. 

43. Use of Discharge Curves for Streams of Unstable Bed. 

44. Methods Available for Measuring Velocity. 

45 . Advantages and Disadvantages of Floats . 

46 . Surface Floats . 

47 . 


Double Floats. 






48. Rod or Tube Pleats . 

49. Details of Float Measurements . 

50 . The Current Meter . 

51. Rating the Meter . 

52. Details of Current Meter Measurements. 

53. The Pitot Tube . 

54. Measurement of Discharge by Chemical Means. 

55. Measurement of Rolative Discharge by Use of Thermometer. 

i 

56 . Computation of Discharge by Means of Slope Formulae . 

57. Accuracy of Stream Gagings . 

58. Time Required for Gagings . 

59. Gagings in Ice-Covered Channels. 






HYDRAULIC MEASUREMENTS . 


1 


1. Introductory . Measurements of the volume of flowing 
water are frequently required in investigations of the amount ob¬ 
tainable from streams for "the public supplies of cities, for power 
development, for irrigation, or for navigation canals; in its dis¬ 
tribution from pipes or canals among the different users; in plan¬ 
ning river improvements; in tests of hydraulic machinery; and for 
many other purposes . The mode of measurement to be employed is 
determined by the quantity of water to be dealt with; by the nature 
of the channel, which may be a pipe, aqueduct, sower, canal, or 
natural stream; by the degree of accuracy required; by the ex¬ 
penditure of time or money involved; and by other circumstances 
peculiar to each case. 

A very small flow may sometimes be conveniently received and 
measured in vessels of known capacit 3 ^; or in vessels arranged upon 
scales, thus permitting the volume to be found from the weight. 

For medium quantities resort must usually be had to less direct 
methods involving the use of formulae with experimentally-dstormined 
coefficients, as in the case of the standard orifice, weir, Venturi 
meter, nozzle, or Pitot tube, which devices, with the accompanying 
theories, will be studied in theoretical Hydraulics. 

Large; volumes flowing in open channels generally require the 
use of floats or current meters, and it is mainly to measurements 
by such means that the following notes will be.confined. 

2 . Relation between D is charge, Mean Velocity and Area . The 
purpose in gaging a stream is to determine with reasonable approx¬ 
imation its discharge, that is to say, the ‘volume passing a given 
cross-section in a unit of time . 

If Q = discharge, in cubic feet oer second, often called 
”second-f39t”, 

A as area of cross section, in square feet, 

V = mean velocity normal to cross-section, in feet per 
second, 

then Q = AV . 

The area of cross-sect ion can be determined from soundings; 
but the mean velocity for the cross-section as a whole can seldom 
be directly observed with satisfactory accuracy and, consequently, 
a closer approximation is usually sought by conceiving the section 
subdivided in a regular way into partial areas, most often vertical 
strips, for each of which in turn its mean velocity can be found 
with tolerable closeness . 

If a = a partial area, in square feet, 

v ss the mean velocity past that area, in feet per second, 
then av = the corresponding partial discharge, 

Q, « £av s the entire discharge, 

and finally V = Q . 

A 








2 


3 . Choice of Location for Discharge Measurements . The place 
of measurement may he closely fixed by special requirements, but if 
not, a wise choice of site will have much to do with the accuracy 
of the results, and with the convenience and cost of securing them. 
A proper selection is therefore very important. 

It is desirable that the cross-sect Ion have a fairly smooth 
and regular outline, and that it be located upon a straight reach, 
the longer the better, of tolerably constant width and depth. 

Such conditions favor the accurate determination of the area; 
promote freedom from the cross currents and eddies which result 
from curves, irregular banks, bowlders, ledges, snags, piers, 
and other obstructions; and tend to ensure that the velocities 
observed be normal to the cross-section. 

In a large proportion of measurements the mean velocity past 
each of a series of verticals in the cross-sect ion is obtained by 
observing the velocity past one point in each vertical, or perhaps 
two or three points, and assuming that the mean for the vertical 
can be derived from such observations by assuming certain common 
relations to hold between them; but, in order that it shall be 
safe to assume 3uch relations, it is essential that the site 
possess the regular features above mentioned. 

If a river station is to be a permanent one, gagings will 
be made at various stages from low to high water, and the rela¬ 
tion thus determined between discharge and gage height; but, in 
order that this relation shalL be constant, it is necessary 
that the bed and banks be stable, and that the section be free 
from irregular back water, such as might result if it were with¬ 
in the influence of varying draughts from a mill-pond below, or 
of temporary obstructions such as ice or log jams, or of tides, 
or of set-back from a stream in flood entering below the site . 

A location above rapids gives an approach to the favorable con¬ 
ditions above a measuring weir and is advantageous; as also is 
one below any lake or extensive stretch of slack water that acts 
as an equalizer of the stream flow. 

Since, at a permanent station gagings will be required both 
at high and at low water, it is important that the site be suf¬ 
ficiently favorable to accurate measurements in both stages. 

It might be suitable for high-water gagings, but very poor for 
those at low stage, on account of sluggish current or partial 
dead water at the latter time . It is therefore safer to make 
a selection in low water than in high, especially since infor¬ 
mation as to discharge in low and medium stages is generally 
more useful than that for high stages. 

It is also desirable that at a permanent station the banks 
be not subject to overflow in high water, or that, if overflowed, 
they at least be free from trees or undergrowth • At velocities 
much below one-half foot per second measurements by current 
meter are apt to be unreliable, and the U. S. Geological Survey 
does not accept a location for a permanent station where the 






3 


velocity is below this limit in more than 15 per cent . of the 
cross-section . 

Ease and safety in making observations are to be considered. 
Gagings from bridges supported on piers are said to be as a rule 
less accurate than those in clear channels; nevertheless, cer¬ 
tain types of bridges are very convenient for operations, and 
because of this and the safe support may well be chosen if the 
piers are parallel to the current . Measurements by wading may be 
practicable in low stages, and from an anchored boat, or a boat 
secured to cross lines in higher water. But when neither wading, 
nor use of a boat nor of a bridge is possible, resort may still 
be had to a permanent cable stretched over the stream, with a 
suspended movable car for the observer, or in a very large river 
to the use of a heavy power-boat . 

The cross-section or sections to be used in gagings should 
be closely normal to the general direction of flow, which may 
be judged roughly from the trend of the banks, but more accurately 
by observing the courses of a number of rod or sub-surface floats. 

4. ga g e * ^ is always well to be able to state definitely 

at what stage of water a gaging was made, and usually a reading 
of the stage at such time is absolutely necessary . It is ob¬ 
viously so in the case of a permanent station where the relation 
between gage height and discharge is to be established, and the 
daily discharge thereafter to be inferred from the daily gage 
readings . It may also be necessary for the following reasons: 

The cross-sectional area, A, to be used in the formula, Q = A V, 
must have a definite value corresponding to the average water 
level prevailing during the taking of observations for V, which 
observations may or may not be made at the same time as the 
soundings from which A is to be found. If made at a different 
time and stage, adjustment of the resulting area to that re¬ 
quired in the computation for Q, can evidently be made only 
from a knowledge of the respective positions of the water sur¬ 
face, as given by gage readings. Again, during the taking of 
soundings in a wide stream the water level may materially change, 
and the soundings can be adjusted to a common level only by 
means of the corresponding gage readings. Finally, the area 
determined from soundings made at a particular stage, supple¬ 
mented if necessary by levels carried up the banks, may after¬ 
ward be adjusted to any other stage, without the taking of new 
soundings, if corresponding gage readings have bean made. The 
change in area is simply that of a horizontal strip, of height 
equal to the difference in gage readings. 

A gage should, therefore, be established at or near the 
place of measurements, and read at suitable intervals during 
the progress of either soundings or velocity observations, to 
the nearest tenth or hundredth of a foot, as may seem justi¬ 
fiable. In the case of the largest rivers two gages are needed, 
one at each shore, as a cross wind may produce noticeable dif¬ 
ference of level within the width of the stream. 

































To permit precision in readings when the water surface is 
roughened by wind or otherwise , the gage may bet enclosed in a 
"stilling box" within which the water surface will be smooth. 

In gagings on the Niagara river at Buffalo a box 7 inches 
square and 7 feet long, with closed bottom and three -J-inch 
holes on the river side, was used, the gage itself being a 
graduated rod supported by a float within the box. Similar 
in principle is the use of a small side basin close to the 
stream, with a pipe or other connection between them. The 
surface in such a basin may be further stilled, if necessary, 
by a floating piece of plank, or even by pouring on oil. To 
effectively prevent wave action within such enclosures Hoyt 
and Grover state that the aggregate area of openings should 
not exceed one-half of of the cross-sectional area of the 
basin (River Discharge, p.30) . It is believed that the level 
obtained by the above devices is a correct mean between wave 
crest and wave trough . 

The gage should be so graduated and set as to avoid the 
inconvenience of minus readings, which may be accomplishod by 
arranging that the zero of the gage shall be several feet below 
the lowest known low water, or preferably, in permanent channels 
level with the deepest part of the cross-section. 

The following styles of gage are to be noticed:- 

(a) Vertical scale-board, rigidly attached to tree, 
pile, bridge pier, or other stable object. 

(b) Inclined scale-board, firmly secured and conform¬ 
ing to slope of banks, and graduated so that its 
figures shall correspond to the usual vertical 
intervals. Sometimes used where a fixed vertical 
scale is impracticable (Rept. Chf . of Engrs., 
U.S.A., 1891, pp. 3820-1). 

(c) Weight lowered by chain from a bridge to contact 
with water surface when reading is desired (see 

* above reference) . 

At important stations, usually for moderate fluctuations 
of level, a continuous record of stage is sometimes maintained 
by means of a self-registoring gage or fluviograph, the main 
features of which are a simple float, or a float plunger com¬ 
pressing air within a manometer; registering mechanism, with 
revolving drum and clock; and the device for conveying the 
motion of the float to the register. On paper wrapped around 
the drum a record is traced both of time and of corresponding 
water level. (U.S.G.S. Water Supply and Irrigation Paper No. 

95. o. 25; Annales des Fonts et Chaussees, 1886, 2d semestre, 
p. 706). 

5. Soundings . It is necessary to determine the outline 
of the cross-section in which velocities are measured in suf- 






































































































































































































5 


ficiest detail sc that whet, in computing the discharge, the 
section is divided into imaginary vertical strips, the area of 
each strip can he calculated with proper accuracy. ?cr this 
purpose soundings are taken. The distances between these, and 
the closeness with which depths are recorded, should he con¬ 
sistent with the evenness and firmness of the river hed, and 
with the general accuracy sought, or probably attainable, in 
the gaging as a whole . It is well to maks the intervals equal, 
except where intermediate soundings may seem necessary to show 
Important irregularities in the hed. A safe principle to follow 
is to seek so to space the soundings that undue error shall net 
result from assuming^the profile a straight line from one to 
another. The general rule adopted by the U. S. Seological 
Surrey is to take soundings as follcws:- 

?or streams from 5 to 10 ft. wide, at intervals of 1 ft. 

■ 10 to 30 • • 2 to 5 ft . 

• 30 to 100 • • 5 to 10 " 

■ ever 100 ■ " 10 to 25 " 

For the largest rivers the intervals should doubtless be still 
greater . 

Depths will ordinarily be read, directly or by estimate, to 

tenths or half-tenths of a foot; but in artificial channels with 

smooth linings readings to hundredths may he appropriate. In 

the violent currents often accompanying floods soundings may 

be difficult or impracticable, and better results obtained, if 

bed and hanks are stable, by correcting soundings observed in 

low stages . Current meter gagings require but a single cross- 

section to be used, and it is a common practice to take soundings 

in each meter vertical, with intermediate ones if needed, the 

meter weight and supporting cord often serving as sounding 

aonaratus . 

* 


The following types of equipment for sounding are employed: - 

(a) Graduated rod, with flat shoe for soft bottoms. 
Suitable for moderate depths . 

(b) For considerable depths, a lead or iron weight, 
ranging from 5 to 40 lbs., according to depth and 
swiftness of current, with line. Flat weight ad¬ 
vantageous for soft bottom, but longer cylinder 
otherwise best and should have conical cavity in 
bottom, to be filled with tallow if samples of 
bed are desired. For heavy weights a reel is use¬ 
ful, and depth may then be found by counting num¬ 
ber of turns of reel in raising lead (See Kept. 

Chf . of Engrs., U.S.A., 1892, p.3119) . 

For the line is variously used:- 

(1) Italian hemp, 1/4” to 3/8” diameter for heavy 
weights, well stretched, tagged, and regular¬ 
ly tested before and after use. Sea grass. 





6 


braided cotton, etc., also in use. Testing 
of line very important; corrections as large 
as 12 feet in 70 feet have been found neces¬ 
sary in lines used in important work. 

(2) Piano or other wire, metallic ribbon, or 

chain, graduations indicated by brazed marks, 
tags, or otherwise. 

In gagings with floats, the value of V in the formula, 

Q, = A V, is presumed to be the mean for the length of the meas¬ 
ured course over which tho floats run, ranging from 200 or 300 
feet downward, according to size and velocity of stream, and 
clearly A must be the mean area for the same distance . In this 
case it is therefore necessary to take soundings in detail not 
only upon the upper and lower cross-sections, but also upon 
sufficient intermediate ones to determine the mean area with 
appropriate accuracy. 

Soundings must be located when taken so that their position 
may be known and, if necessary, plotted upon the profile of the 
cross-section. The following methods of location are applicable, 
according to circumstances:- 

(a) Distance from bank, or from reference line on bank, 
read on chain, tape, or graduated line, stretched 
across the stream. Suited to channels of moderate 
width , 

(b) Distances read by stadia from shore end of range. 

(c) Angles from base line on shore to sounding stations 
measured with transit . 

(d) Sextant or goniograph angles to known points on shore 
observed from boat . 

(e) Boat brought to intersection of range lines previously 
established on shore . Common on very large rivers 
where it is desired to take repeated observations at 
given points. (See Rapt. Chf. of Engrs., U.S.A., 

1891, p.3532; also Johnson’s "Surveying", p.289). 

When the stream surface is not affected by & cross wind 
there is probably no material error, at a properly located 
gaging station on any river, in assuming the water line of the 
cross-section to be level. Even when the surface is smooth, 
however, there is evidence that the line is not in all cases 
strictly level. In gagings below the Vyrnwy reservoir, the 
surface at the center of the stream (width 40’) was found 
slightly lower than at the sides in certain stages. During 
gagings of the Niagara river at Buffalo (width 1800’) the water 
surface at the center was about 0 *2 foot lower than at the side . 
On the lower Mississippi river the reversal of the current from 






















































7 


ona bank to the other at bends causes a difference of level 
between opposite banks sometimes as great as a foot, and gives 
the^river in the vicinity a warped surface. Some observations 
on large streams have seemed to show a convex surface during 
a rising stage, plane during steady flow, and concave during 
a falling stage, but evidence as to this is not conclusive. 

6• Interpolation of Soundings . If soundings have been 
taken at irregular intervals in the cross-section, and depths 
be desired at regular intervals for puri>oses of computation 
or otherwise, these may be obtained by direct interpolation 
upon the profile, assuming it either straight between observed 
soundings,or curved, as may be thought best. 

It sometimes happens in the case of a river with shifting 
bed that soundings are repeated at the same points after pn 
interval cf some days, or even weeks, 
and that at an intermediate date ve¬ 
locities are observed for the purpose 
of ascertaining the discharge . The 
cross-section must be estimated as 
well as possible for the time when 
velocities were measured. If the 
conditions are such that the scour, 
or fill, in the bed may fairly be 
assumed to have progressed uniformly 
between the successive dates of 
sounding, and gage heights have been 
read, the observed soundings may be 
corrected to their probable value at 
the intermediate date as follows:- 

Sketch a figure or which depths 
and time intervals are conceived as 
laid off to scale (Fig. 1). 

Let d* and d" = depths as observed at the same station on 

two different dates. 

x = required depth at intermediate date . 

h f , h M and h = corresponding gage heights. 

m = time interval between h* and h. 
n = time interval between h* and h M . 

Then x may easily be found from the geometrical relations 
of the figure . 

7. Computation of Area from Soundings in a Single Cross - 
Section . As a rule, in computing discharge the area ofthe 
cross-section is needed in detail for each of successive ver¬ 
tical strips into which it is conceived to be divided. Usually 
the strips are of equal width, and either lie between succes¬ 
sive observed soundings or have such soundings at their centers . 
On any plotted cross-section, however, strips may be arbitra¬ 
rily laid out independently of observed soundings, if there is 
any call for doing sc, and the accompanying depths scaled or 
numerically interpolated between soundings . Generally the bed 


// 




A 

i 


A 

i 

| 

l 

1 

1 

1 . 

1 

1 

JC 

I 

ct* 


h 
h 

h' 
























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8 


profile is assumed as a series of straight linos extending from 
one sounding to the next; sometimes, and more accurately, as a 
series of parabolic arcs, convex downward. Perhaps most often 
the strips are combined singly with their corresponding veloci¬ 
ties in computing the discharge; y/hii© at other times, to 
shorten somewhat the labor of computation, double strips are 
taken . The problem then is to find v/ith reasonable approxima¬ 
tion the areas of the strips. 


If a * area of a strip in 

square feet, 
b = its width in feet, 
x = its mean depth in feet, 
then a = bx. 

In Pig. 2 let c, d, e represent 
successive observed or scaled depths, 
e qu ally spaced . 

(1) Area of any strip may be 
measured by plan line ter. 

( 2 ) For any strips, as f g j 1, 
regarded as a trapezoid, a = b 



(3) For cross-hatched strip, regarding d as mean depth, 

a = bd 


(4) For cross-hatched strip, regarding i j and j k as 
straight lines, 

a = b £JLli-LJ° 

8 

which may also be written 


a = b d + (c-d) + (e-d ) 

8 

(5) For double strip, regarding each single strip as a 

trapezoid, ^ ^ c , Me , . 

4 

(6) For double strip, more accurately, regarding i j k as 

a parabolic arc, c + 4d + «. 


a = 2b 


8 


At each end of a cross-section there is frequently & strip 
of odd width or shape, the area of which may be estimated by 
whatever method commends itself to the judgment . 


8 . Computation of Mean Area from Soundings in Several 
Cross-Sections . As noticed in Art . 5, "TFTs ne~ce~s~sary~"in the 
case of™gagings v/ith floats to find the mean cross-section for 























9 


the distance covered by the 
float runs . In Pig 3 let BB 
and B’B' represent the water’s 
edge; C N a base line on shore; 
C C’ and N N’ , at right, angles 
to the base line, cross ranges 
fixing the length of run; and 
3) D’ etc . intermediate equally- 
spaced cross ranges, if such be 
required for the proper deter¬ 
mination of the mean section. 
Th® number of these will depend 
upon the regularity of the bed, 
the length of the course, and 
the general accuracy to be ex¬ 
pected in the gaging as a 
whole . Soundings are taken at 
the same distances out from the 
base on each cross range, or if 
not may be interpolated at such 
distances . 


s' 


c 

n 


F 

G 



0 ' 


AS 


i 

h$- 

1 

<0 




The mean cross-section may now be constructed, as shown, 
from a seriee of ordinates corresponding to the mean depths 
along successive longitudinal sections, such as a-n . The mean 
depth of any such longitudinal section, shown at the right, 
may be found from the numerical mean of depths at the inter¬ 
sected cross ranges or, if greater accuracy be required, by 
computing the area of the section and dividing by its length. 
The mean distance out from base line to water’s edge may be 
similarly found, and the representative cross-section is then 
complete . 


9. Complex Motion of Plowing Water . In a general se , 
and approximately, the flow in a river or other open channel 
may be "steady”, and it may be allowable, as a matter of con¬ 
venience, to speak of particles of water moving in "stream 
lines", and of "threads" of velocity; but careful observers 
are agreed that, except perhaps when the velocity is exceeding¬ 
ly’ low, the actual motion of particles, even when the stream 
appears to the eye entirely tranquil, is extremeiy unsteady and 
complex. The velocity is not really constant at any point, but 
everywhere- varies from instant to instant . Innumerable whirl¬ 
pool© are formed, and solid particles in suspension have a 
rotary motion, both horizontally and vertically, which inter¬ 
feres materially with their settling to the bottom. The vari¬ 
ability in direction is shown by this confused motion of par¬ 
ticles of silt, by the irregular paths often taken by floating 
chips, by the swaying of v/eeds, etc. The paths of individual 
particles are evidently’ interlaced irregularly from one moment 
to another in the most intricate manner. 

The "thread" of maximum velocity is never long ir. the same 
place, but constantly sways from side to side and rises and falls. 




























































10 


In the Mississippi river at Arkansas City it has been known to 
move laterally 1800 feet in one day, and half that distance be¬ 
tween morning and afternoon. Leviavsky, studying with floats 
the currents near the surface in the river Dnieper, near Kieff, 
found no portion of the river where all motions were parallel; 
there were points on the ?/ater surface toward which floating 
bodies tended from both sides, and others at which if two 
floats were started close together they would invariably sep¬ 
arate within a certain distance (Eng. News. Sept. 1, '04, p.184). 
It is plain, then, because of the uncertain motions, that veloc¬ 
ities must be measured at numerous points in the cross-section, 
and that even then the difficulties may be considerable in the 
way of securing a close approximation to the average velocity 
past the section at any giver? time. 

10. Velocity Pulsations . Even while the wetted cross- 
section and discharge of a stream, and consequently its mean 
velocity, remain substantially unchanged, i .e ., while the flow 
remains technically "steady", it is a matter of common observa¬ 
tion that the forward velocity at any particular point is sub¬ 
ject to constant pulsations, often of considerable magnitude. 
These pulsations, which sometimes mean a variation of velocity 
amounting to 50 per cent, of the greater value within one minute, 
become perceptible to the eye in the behavior of floats, and to 
the ear in the varying intervals between the taps or buzzes in¬ 
dicating revolutions of the electrical current meter. Cunning¬ 
ham found in the Ganges canal a number of similar floats, run 
in rapid succession over the same course, to show commonly a 
range in velocity of as much as 20 per cent . of the mean. 

Prancis observed tube floats, run in a canal at Lowell, under 
apparently identical conditions, to vary in velocity from about 
8 \ per cent, above the mean to about 11-J- per cent, below it. 
Marr*s measurements in the Mississippi at Burlington showed the 
velocity at 9 feet depth, as given by individual one-mir.ute 
current meter runs, to vary from the mean for 32 minutes by 
from about 10 per cent . above to about 11 per cent, below. 

These pulsations are observed to occur in streams of all 
sizes, and are evidence of an irregular motion in the water 
which has been likened to the unsteady motion of the wind 
shown in the swaying of weather vanes and the fluttering of 
pennons . The cause of the unsteadiness is not well known. 

The surface level of a stream is said also to show usually a 
pulsation covering from 30 to 50 or more seconds, and giving 
a vertical oscillation through several hundredths of an inch 
(■J.S.G.S. Water Supply and Irrigation Papers, No. 56, p .42); 
but Henry's experiments in the St. Clair and Niagara rivers 
indicated that the velocity pulsations were not synchronous 
with oscillations of the surface level, which itself varied 
there very irregularly. Prancis attributed the varying velocity 
of tube floats, as noticed above, to the constant interchange 
of place of currents of different speed. 

Experiments by Marr in the Mississippi, with simultaneous 
observations on several meters, showed the changes in velocity 










11 


to occur at about the same time from surface to bed, but to 
increase in magnitude as the latter was approached. This 
latter law has frequently been noted by others. Harlacher 
observed velocities in the Rhine to vary in a few seconds 20 
per cent, at the surface, and 50 per cent, at the bed. Unwin 
found in the Thames the velocity as measured in individual 
meter runs of 100 revolutions each to vary from the mean given 
by a continuous run of 1600 revolutions by from 

+ 8.3 fo to - 6 .0,€ at one-half metre depth, and 
+ 16 .1 fo' to - 37 .4 *fo at six metres depth . 

Upon plotting times as abscissas and velocities as ordi¬ 
nates it has been shown that curves result displaying two sets 
of waves,- minor ones of from 15 to 60 seconds amplitude, and 
major ones of from 3 to 6 minutes, or even'longer. Unwin found 
for the Thames that, plotting velocities as obtained from suc¬ 
cessive runs of 100 revolutions each he obtained a very irreg¬ 
ular curve, but from successive runs of 500 revolutions each 
very nearly a straight line, the meter making 500 revolutions 
during each of a dozen successive equal periods of time (Min. 
Proc . Inst. Civ. Engrs ., Vol.71, p.4l). 

Prom what has been said it must be seen that the velocity 
pulsations have an important bearing not only upon the selec¬ 
tion of an instrument for measuring velocity, but also upon 
the proper length of observations. A float, for example, may 
move down stream under the influence of a maximum or a minimum 
impulse, and only by chance with the true average velocity, 
which therefore can be learned only by taking the mean given 
by many repetitions. In Major Cunningham's Roorkee hydraulic 
experiments on the Ganges canal it was thought to be demon¬ 
strated that about 50 repetitions with floats were there neces¬ 
sary to obtain a fair average . On the other hand, with a cur¬ 
rent meter, hold at one point, the fluctuations may be averaged 
simply by increasing the duration of the run. 

Baum concluded from his studies in the Rhine that for ac¬ 
curately averaging the pulsations at a point it was necessary 
to take continuous observations for as long as an hour. Gen¬ 
erally, however, it is considered that the longer waves of ve¬ 
locity may be covered and a satisfactory average be obtained by 
continuing a meter run at a given point for from 5 to 10 minutes, 
and this has been a common practice with government engineers 
in this country. Studies by the Hydrographic Department of 
Hungary, in the river Theiss, have indicated that three or four 
maxima and minima may be expected in a period of about 3 minutes 
where the maximum velocity is as high as 2 metres per second, 
and in a period of about 5 minutes in the slow filaments near 
the bed, and the department engineers have therefore adopted 
the rule of making the duration of measurements from 3 to 5 
minutes, according to the speed of current (Annales des Ponts 
et Chaussoes, 1898-3, p .301) . 

Since any single observation with a float, or any short 
observation with a meter, clearly gives but an accidental value 










12 


for velocity, which may even be a maximum or a minimum, it is 
plain that for the purpose of critical comparison of velocities 
at different points only many repetitions with floats or long- 
continued runs with meter can be of much value. To apply such 
a requirement, however, at the many points which it might be 
desirable to occupy in the ordinary gaging of a stream of large 
cross-section would call for an expensive outlay of time, and 
would involve the further serious difficulty that before the 
observations had progressed far a change in stage:, and conse¬ 
quently in the amount and distribution of velocity throughout 
the cross-section, might occur. A compromise has, therefore, 
usually to be accepted between duration and number of observa¬ 
tions, and meter runs of 5 or 10 minutes each are made at a 
moderate number of points, or runs of say 1 minute, more or 
less, are made at a larger number. The shorter the run, the 
greater the necessity of increasing the number of observations 
and of distributing them well over the discharge section. With 
floats, it is common to make single runs at numerous stations 
well distributed across the stream, and then to draw an averag¬ 
ing curve through the velocity points thus determined. 

11. General Law of Variation of Yelocities past a Cross - 
Sect ion . In spite of the intricacy and variability of the 
paths of the individual particles of water, and the pulsations 
in velocity at any given point, there is, for the mass of flow¬ 
ing water as a whole, a general law of distribution of veloci¬ 
ties in the cross-section, upon a knowledge of which are based 
a large proportion of stream gagings. On account of the rough¬ 
ness of the channel lining, giving rise to numberless eddies, 
and the viscosity of the water, a resistance to the flow is de¬ 
veloped next the bed and banks and is felt, in lessening degree, 
as those are receded from; so that in a general way the highest 
velocities are found in the central part of the stream, and 
well toward the surface. It is noticeable, however, that gen¬ 
erally the maximum of the velocities past the different points 
of any vertical line is found, not at the water surface, as 
might be expected, but below it, and sometimes as far down as 
mid-depth . 

The cause of this depression has been a matter of much 
speculation, and as long ago as 1867 Francis made experiments 
in the Northern and Western canals at Lowell, which were thought 
to show that the relatively slow movement at the surface is due 
to eddying masses of slow-moving water near the bed being de¬ 
tached and forced upward to the surface by quicker currents; 
and others have supported substantially the same view. Cunning¬ 
ham, however, considered the air to supply an. efficient resisting 
margin, and the surface velocity to be necessarily thereby re¬ 
tarded. No doubt a strong wind with or against the current 
affects the position of the thread of maximum velocity, but it 
appears that even with a wind blowing down stream the maximum 
velocity may often be found below the surface, and air resistance 
seems an inadequate explanation of the phenomenon in question. 

The theory best agreeing with observed facts appears to 

































































* 














13 


be that elaborated by Stearns (Trans. Am. Soc . C .E ., Vol. 12, 
pp. 331 et soq.) , according to which there is an upward flow 
at the sides of a channel, carrying with it the slow-moving 
water always found in the immediate vicinity of channel linings, 
which upon reaching the surface flows obliquely toward the 
middle of the stream, retarding by its slower movement the 
velocity of the surface layers and constituting thereby the 
main cause of the depression of the thread of maximum velocity 
below the surface . This upward and outward flow was made evi¬ 
dent by placing a vertical projecting board at the side of a 
rectangular canal, when sawdust that had been mixed with the 
water was seen to rise along the up-stream surface of the board 
to the water surface and there to spread out toward the center 
of the channel. Cunningham thought such outward flow well 
demonstrated by the behavior of surface floats in experiments 
in the Ganges canal, it sometimes having been found necessary to 
make 100 runs before 3 could be obtained that followed a fair 
course only a dozen feet long and about 8 inches from a straight 
vertical bank, the floats showing a constant tendency to swerve 
outward. 

Inasmuch as the locus of the thread of mean velocity for 
any vertical line is likely to rise, and fall with that of the 
thread of maximum velocity, it is of much practical importance 
to notice the effects upon the latter which should follow the 
application of Stearns 1 theory, and which seem tc be generally 
realized in practice, other things equal, the locus of the 
thread of maximum velocity, in any vertical line - 

(a) Should be higher, the smoother the channel lining, 
and vice versa. 

(b) Should be higher, the shallower the water section, 
and vice versa; and for a similar reason it should be 
higher, the more gently the bed slopes away from the 
water’s edge, and vice versa. In the central part, 
at least, of relatively shallow streams the maximum 
velocity is apt to be found at or very near the surface . 

(c) Should be higher at the greater distance from the 
sides, and vice versa. In the La Plata river, sev¬ 
eral miles from shore, Revy found the maximum veloc¬ 
ity at the surface, though the water was there 25 
feet deep . 


Pig. 4, showing the results of observations by Darcy, makes 
clear to the eye the general distribution of velocities to be 
expected in a rectangular section, and displays not only curves 
connecting the loci of equal velocities in the cross-section, 











but also curves of variation of velocity in selected vertical 
lines, and others similarly in horizontal lines; they illus¬ 
trate well some of the laws stated above . 


14 



12. Velocity Measurements at Numerous Points Uniformly 
Distributed over the Cross-Section . Since it is impossible 
to measure all the different velocities in a cross-section, 
and since a reasonably close approximation to the mean veloc¬ 
ity and discharge is all that is required, it is apparent that 
if the cross-section can be satisfactorily subdivided into im¬ 
aginary equal elementary areas, and the velocity observed at 
the center of each of these, the numerical mean of th6 observed 
velocities will be, approximately, the desired mean velocity 
for the entire section • 


If 


and 


a ~ the common elementary area, 
n =5 the number of such areas, 

A ™ the entire cross-sectional area, 

v = the observed velocity at center of each elementary 
V = the mean velocity for the entire cross-section, 

Q ss the total discharge; 


area, 


then V = JLI> 
n 


and Q * A V = Z av, 


This method can successfully be applied to artificial sections 




































15 


of simple geometrical shape, in which the flow is sufficiently 
steady not to change materially during the time required for 
the "point measurements", and a high degree of accuracy may he 
attained, say within of the truth (e.g. see Watkins gaging 
in old Croton aqueduct, mentioned by Freeman in Report on New 
York's Water Supply, p.i ss) 

13. Reduction Coefficients . On account of the great 
time and expense that would he necessary for gagings in large 
channels if based simply upon a wide distribution of point 
measurements, and because of the danger of a total change in 
the conditions of flow during a lengthy operation, very exten¬ 
sive studies have been conducted in the effort to discover 
such relations between the various velocities in the cross-sec¬ 
tion as would permit of reducing to a minimum the necessary 
number of observation points; these being so chosen as either 
to give directly certain desired mean velocities, or to give 
them through the application of "reduction coefficients" to 
the observed values. (In the Roorkee hydraulic experiments 
about 50,000 velocity measurements were made with this general 
aim in view) . Such relations have been sufficiently well de¬ 
termined to ensure under favorable conditions a fair approxi¬ 
mation to the discharge, and yet it is true that they are £o 
affected by differences of width, depth, velocity, or channel 
lining, that as the number of observation points is diminished 
the likelihood of close accuracy in the computed result is 
correspondingly lessened. 

14 . Velocity Measurements at but One Point in the Cross - 
Section . There appears to be no point in the cross-sect-icn 
the velocity at which bears, under varying conditions, a suf¬ 
ficiently constant relation to the mean velocity for the entire 
section to ensure the determination of the latter velocity with 
closeness from observations at the single-point. Nevertheless, 
by measuring either the central surface velocity, in artificial 
channels, or the maximum surface velocity, in natural channels, 
and applying a mean reduction coefficient of say 0.8, a rough 
approximation to the discharge can be made, with an error per¬ 
haps not likely to exceed 10^; and by experimentally determin¬ 
ing a special coefficient for the particular cross-secticn used 
subsequent measurements could doubtless be made with much 
greater accuracy. This method may be in order in a reconnoie- 
sance , ?/here great accurac;/ is not essential; or in hasty meas¬ 
urements of flood discharge, where, also, close accuracy is 
often not needed . 

Tf Q, = discharge, 

A = entire cross-section, 

V = mean velocity past entire cross-section, 

V g = observed surface velocity, 

C = reduction coefficient, = ~ , 

then Q = AC V s . 

Measurement of the central surface velocity was regularly 














16 


practised in the early history of gagings in certain water-power 
canals at Lowell, Mass., hy finding the velocity of small sur¬ 
face floats at mid-stream, and using reduction coefficients 
which were independently determined at 0.84? for the Western 
canal (width about 2? feet, depth about 8 feet, mean velocity 
about 2.7 feet per second); and at 0.814 for the Merrimack 
canal (width about 30 feet, depth about 8 l/2 feet, mean ve¬ 
locity about 1.7 feet per second); but the uncertainty as to 
whether reliable results could be obtained by the same method 
for other canals of the system presenting less favorable condi¬ 
tions led to its entire abandonment •• 

The numerical value of this coefficient appears to range, 
under different conditions, between 0.75 and 0.95. In extensive 
gagings below the Vyrnwy reservoir (stream bed 32 feet wide, 
bank slopes 2 base to 1 vertical) it was found to range between 
0.78 and 0.94, with an average of 0.83. It seemed there to be 
independent of depth and velocity, but a study by Prony of ex¬ 
periments by Du Buat in small wooden troughs led him to the 
following formula in which the coefficient is seen to increase 
with the velocity:- 

V v 7 .78 

C =y = —£2-1.-— (Lowell Hydraulic Experiments, p.154), 

03 V cs + 10.35 

V C8 representing the central surface velocity. 

For observed surface velocities ranging from 1 to 10 feet 

per second this formula will be seen to give values of the co¬ 

efficient ranging from 0.77 to 0.87. 

It is urged, however, that the surface velocity at any 
particular point, such as mid-stream, is easily affected by wind 
and is otherwise very variable, and that better results will be 
obtained by observing the maximum surface velocity, which may 
or may not occur at mid stream; in the Vyrnwy experiments its 
position was found not to be constant, but to be generally 
within a few feet of the center of the channel, on either side . 
Under this plan several comparative observations must evidently 
be made to locate the point of maximum velocity. Prom a study 
of 24 different gagings, including both small and large streams. 
Prof, von Wagner found the value of the coefficient for this 
method to be given by the formula 

C ~ Vmi ~ + 0 "03. V ms , 

V m representing the maximum surface velocity in the whole 
width of the stream. (Min. Proc. Inst. Civ, Engrs., Vol.71, p.92). 

Por velocities of from 1 to 10 feet per second this formula 
gives a range of value for C from about 0.71 to about 0.81; when 
applied to the Vyrnwy gagings it was found to agree well with 
observed results, though usually giving somewhat too large a 
value, the error being in 20 cases out of 33 less than 3in 













17 


29 out of 33 less than 10^, and in no case as great as 16$. 

It will he noticed that for a straight channel, of symmet¬ 
rical section and approximately geometrical shape, in which 
case the center surface velocity is likely to he also the max¬ 
imum surface velocity, or very nearly so, von Wagner's formula 
gives for ordinary velocities a coefficient lower than Prony's 
hy say 0.06 . On the whole it seems fair to suppose that for 
conditions (noticed in Art . 11) tending to depress the locus 
of maximum velocity in a vertical line, such for example as a 
relatively narrow or deep canal with vertical sides, Prony's 
formula will apply best, and in a natural river section von 
Wagner's . 

Thus, in an artificial canal of say 800 square feet crcss- 
section and an observed surface velocity at mid-stream of 4 
feet per second, the discharge is likely to lie between 

800 x 0.77 x 4 = say 2460 cubic feet per second, 

and 800 x 0 .87 x 4 - say 2780 cubic feet per second; 

while in a natural river channel of 800 square feet cross-sec¬ 
tion and an observed maximum surface velocity of 4 feet per 
second, the discharge~~ITTikely to lie between 

800 x 0.71 x 4 = say 2270 cubic feet per second, 

and 800 x 0.81 x 4 = say 2590 cubic feet per second. 

It has also been suggested that instead of surface veloc¬ 
ity the central mid-depth velocity be observed, as being less 
variable than the former (in the Vyrnwy gagings the corresponding 
reduction coefficient ranged from about 0.83 to about 0.93, with 
a mean of 0.89); or, still better, that the mean velocity past 
the center vertical be observed (in the Vyrnwy gagings the co¬ 
efficient for this case ranged from 0.87 to 0.99, with a mean 
of about 0.93) . Such measurements would he less simple to make, 
however, than those on the surface, and with a stream in flood 
might be impracticable . 

Still another method of quick approximate measurement has 
been proposed by Unwin, based on the fact that there must be 
in every cross-section two verticals (at least) past each of 
which the mean velocity is equal to the mean velocity for the 
entire section. A determination of that value in either ver¬ 
tical, or better in both, means for which will be described in 
a later article, would enable the discharge to be directly com¬ 
puted. It is hardly to be supposed that the location of either 
of these verticals could be predicted in general for irregular 
cross-sections, but for artificial sections they appear to come 
at about one-third the stream width each way from the center. 
Applied to 27 sets of Vyrnwy gagings this method gave the dis¬ 
charge for 8 sets within 1 ft, and for 24 sets within 5^. 

15 . Velocity Measurements in Each of a Series of Verticals . 
The great majority of stream gagings are made by observations 
in a series of vertical lines distributed across the stream, 
the immediate purpose being to obtain for each vertical the 















































• 








• 



























































































































































































18 


mean of all the varying velocities past it from surface to bed. 
From these mean velocities the discharge can then he computed 
hy methods to he explained later. The number of observations 
in each vertical is limited to as few as can he relied upon to 
give with satisfactory accuracy the mean velocity past the ver¬ 
tical, which is computed hy assuming to hold a certain relation 
between the different velocities and applying, if necessary, 
appropriate reduction coefficients determined hy independent 
observations. The law of distribution of velocities in a ver¬ 
tical assumes, therefore, prime importance, and an enormous 
amount of experimentation and study has been given to it . 

16. Vertical Velocity Curves . If a stream be in techni¬ 
cally steady flow, and in any vertical line the velocities past 
all points be conceived as simultaneously measured, the obser¬ 
vation being long enough to average the pulsations, and the 
velocities be then laid off to scale from a vertical line, a 
curve will result which for brevity is called the "vertical 
velocity curve". The approximate geometrical shape of this 
curve is of some consequence, since the known relations hold¬ 
ing for a given geometrical curve may suggest the number and 
desirable depths of observation points, and may lead to logical 
and simple rules for computing the mean velocity. The determi¬ 
nation of the numerical values of constants in the equation of 
the curve is of minor importance, but if attempted can best be 
effected by the method of least squares. 

Extensive observations for the express purpose of determin¬ 
ing the representative shape of the vertical velocity curve have 
been made with double floats, but more frequently with meters, 
two methods being used with the latter:- 

1st, and most commonly:- A single meter is held suc¬ 
cessively at different depths in the vertical, usually at 
each tenth of the total depth, and at each point long 
enough to average the effects of pulsations. This method 
is likely to require at least an hour for a single verti¬ 
cal, during which time an important change may occur in 
the mean velocity itself . 

2nd:- A number of meters are used simultaneously 
if possible enough to occupy all the desired points in a 
vertical and thus give the entire curve at one observa¬ 
tion . On the St . Clair river eleven meters were used by 
Haskell at one time, runs of ten minutes being made with 
the meters placed at each tenth of the depth. 

A single set of observations in a vertical, especially if 
obtained from floats, or from short runs with a meter, may give 
a very irregular curve (Fig. 5), which, however, will become 
more and more regular as the number of sets is increased and 
averaged; and it is only from many sets combined that a typical 
curve results from which geometrical properties can properly 
be drawn (Fig. 6). The combining, for purposes of general 

(see p. 20) 





19 



PADUCAH STATION MEANS 



i o y iv 

23.8 13.0 17.3 12.0 _1!M)_ 17.0 



r 





T" 


“ 7 



T" 


IT 







-t 


r 



L 


- 1- 

- 

) 


-f 



i ~ 


-f 



7 : 


IT 

7 





±1 



J . 

_L 


r 


J- 



HIGH WATER LOW WATER 


MEAN OF 12 VERTICALS MEAN OF 22 VERTICALS 



F/p.S. Fert/ca/ Fe/ocftc/Curves ojbta/neft jb// 
/ff/ss. ft/ver Camm/s-s/on c/f Pnctucah. 


Pee fper eeconft 

_A 6 _ AS_ __2..0 

Plater Surfer cte 



F/g, 6. Ferf/ca/ Ve/oc/tj/ Curve . 
Connect/cutft/ver of TPompsonv/Z/e. 
Gr&rrcf/Wean tram£///s’0/beervat/tens. 


(Pig, 5, The stations in the cross-section are given by 
number, in order from the left bank, and the water depths at 
those stations. The upper row shows a few selected curves 
given by single sets of observations. Then follow curves 
obtained by combining and averaging for each station all the 
observations of a year, succeeded by a combination curvs for 
all the stations taken together, with separate mean curves 
also for high water and low water conditions, respectively. 
See Starling on The Discharge of the Mississippi River; 

Trans. Amer. Soc. of Civ. Engrs., Vol. 34, p. 378.) 



































































































































































































20 


study of form, of vertical velocity curves obtained under vary¬ 
ing conditions of depth and of mean velocity, or of position 
in the cross-section, is usually effected by determining for 
each individual curve its mean velocity and the relation of 
the velocity at each tenth of the depth to that mean, and then 
averaging the relative values thus obtained from all the dif¬ 
ferent curves which are to be combined. Since, however, the 
shape of individual curves may vary more or less with the dif¬ 
ferences in the conditions just mentioned, it is open to ques¬ 
tion whether averaging may properly be attempted of curves for 
which those conditions are notably different . 

There has been much diversity of opinion as to the geomet¬ 
rical curve best representing the distribution of velocities 
in the vertical, and investigators have variously found it to 
be the ordinary parabola with horizontal axis, a parabola with 
vertical axis and vertex at or below the bed, an ellipse with 
horizontal minor axis below the water surface, an hyperbola, 
a right line or a broken right line, a logarithmic curve, etc. 
Each of these may perhaps best fit some particular set of ex¬ 
periments, and in some cases two or more curves might about 
equally well fit the same sot; but the most extensive series 
of observations, such as those of Humphreys and Abbot, Ellis, 
Cunningham and others, have led their authors to adopt the 
common parabola with horizontal axis as the typical curve, and 
this is now very generally assumed. Prof, von Wagner, from 
the investigation of 64 curves of small streams and large rivers, 
thought the law of distribution most accurately represented by 
a compound curve (Pig. 7; see Min. Proc . Inst. Civ. Kngrs., 

Vol. 71, p. 89),- a parabolic arc extending from the axis, or 
line of maximum velocity (which he found to lie variously from 
0 to nearly 0.3 of the total depth below the surface) down to 
about 0.8 of the total depth, but above and below this inter¬ 
val deviating inward from the parabola, with zero velocity at 
the bed. 











21 


17. Determination of Reduction Coefficients for Vertical 

Velocity Curves. The common necessity, on account of fluctua- 

tions or stage, of limiting the duration of any single gaging 
to a moderate time, say one or two hours, results in the gen¬ 
eral practice for large streams of taking observations at but 
one point in each vertical and, if necessary, applying a co¬ 
efficient to reduce the observed velocity to the probable mean 
velocity past the vortical. 

Although approximate values are known for the coefficient 
for all relative depths in the vertical likely to be used, it 
is desirable, in order to get the most accurate values, to make 
at each permanent gaging station a reasonable number of inde¬ 
pendent determinations of the coefficient by means of complete 
vertical curves observed at a number of different locations 
in the cros3-83ction, and at different stages. The work can 
be done with one meter, but better and more rapidly with two. 

In the latter case one meter is held steadily at the relative 
depth proposed to be occupied id routine work, called the index 
point, and the other is held for five minutes, more or less, at 
each of the other points in the vertical (say at each tenth of 
the depth) necessary for a good determination of the vertical 
velocity curve. The practice of the Hungarian Hydrographic 
Department for deep streams has been to take observations for 
vertical velocity curves as follows: as nearly as possible at 
the water surface; then at depths of 0.25, 0.75, 1.50, 2.50, 

4.00, and 6.00 meters; then at intervals of 2.00 meters and, 
finally, of 1.50, 1.00 and 0.50 meters; and, at the lowest 

point, as near as possible to the bed (Annales des Ponts et 
Chaussees, 1898-3 - p.301). Prom the complete curve, the 
ratio of the mean velocity in the entire vertical to the veloc¬ 
ity at the index point is easily obtained. 

18. Choice of Observation Points in Verticals. If the 
relations between the different velocities in a vertical ware 
invariable, it is plain that we might measure the velocity at 
any convenient point in the vertical and from it determine the 
mean velocity from surface to bed. But the relation of the 
velocities at the different points to the mean for the vertical 
is not invariable, and varies more for some points than for 
others; moreover, the difficulty of accurately locating the 
observing instrument, and of making an accurate measurement 
with it when in place, is considerable at large depths in 
swift currents; and in time of flood, when there is much 
floating debris, a measurement much below the surface may be 
entirely impracticable. Consequently there is call for the 
exercise of judgment in choosing the point or points at which 
observations shall be made, and the following considerations 
are important:- 

(a) In large and deep streams the vertical velocity 
curve is generally more nearly vertical from the 
surface to mid-depth than farther down, and failure 
to place the meter or floats exactly at the intended 








22 


depth is therefor© likely to introduce leas error into 
the result at or above mi&-depth than below it. In 
gagings of the Niagara river at Buffalo in 1897-*98 
all meter observations were made at 3/l0 depth. 

(b) From surface to bed the pulsations in velocity in¬ 
crease (Art. 10), and the error due to these is there¬ 
fore less, for a short observation, at the higher 
levels. 

(c) The ratio of the velocity at any particular depth in 

a vertical to the mean velocity for the vertical seems 
to be most constant in the vicinity of mid-depth, say 
in the middle third of the depth, and least constant 
near the surface and bed, especially in the upper and 
lower tenth of the depth. The error, therefore, in 
individual verticals, and in individual gagings when 
based upon a small number of verticals, due to devia¬ 
tion from the assumed ratio, is likely to be less if 
observations are made in the middle portion of the 
vertical than if above or below. Humphreys and 
Abbot in their extensive observations in the Missis¬ 
sippi found the constancy greatest at mid-depth. 
(Report upon the Physics and Hydraulics of the 
Mississippi River, p. 311). A study of mean velocity 
curves for seven streams considered in connection 
with the water supply of New York City showed the 
least variation between them at 6/l0 depth (U. S. 
Geological Survey Water Supply and Irrigation Paper 
No. 76, p. 45). 

(d) The small Price meter, at least, does not give accu¬ 
rate results, at velocities above 1.5 feet per second, 
if held nearer the water surface than one foot. 

19. Velocity Measurements at or near Surface. The ratio 
of velocity at surface to mean velocity in a vertical shows less 
constancy than the ratio for almost any other position in the 
vertical, and surface measurements offer a corresponding dis¬ 
advantage for accurate gagings. Still, the deviation is not 
excessive, and for stream reconnoissance or flood measurements 
the approximation obtained may be quite satisfactory. Further¬ 
more, the greater ease and certainty in manipulation of.the 
meter near the surface in swift currents, the opportunity for 
quickly raising it out of the water, if necessary for its pro¬ 
tection from floating debris, and in case use of the meter is 
impracticable the easy substitution of surface floats, often 
make this method a convenient and even a necessary resort. 

Its use for flood gagings was advocated by M, Charles 
Ritter in an article in Annales des Ponts et ChausseaB (1886-2 - 
pp. 697, et seq). For measuring velocity he used a modified 
Darcy tube, and considered that current meter observations by 
Baumgarten, Harlacher and others, had shown that a reduction 








23 


coeffic 3nt of 0,85 applied to the velocity as measured say 
from 4 inches to 6 inches below the water surface would give 
the mean velocity in the vertical with an error not likely to 
exceed 5J?; but in order to secure a safe or superior value for 
the discharge he proposed to use, not 0.85, but 0.90, and be¬ 
lieved that the result would generally be in error less than 

lotf. 


The advantage of surface measurements in high stages of 
streams, when the rapid fluctuations preclude the slower but 
more accurate method of observation at several points in each 
vertical, was also pointed out by Harlacher and Richter (Min. 
Proc. Inst. Civ. Engrs., Vol, 91, pp. 397 et seq). For 28 
series of verticals obtained in gagings in the Danube and in 
Bohemian rivers, comprising about 300 individual curves, they 
found the average value of the reduction coefficient 0.84, 
with a range for the different series (not for individual 
curves) from 0.79 to 0.91. They quoted also similar observa¬ 
tions by Swiss and Dutch engineers showing mean values for the 
coefficient of•0.835 and 0.87, respectively. 

Hoyt and Grover give figures for between 800 and 900 ver¬ 
tical velocity curves measured by engineers of the U. S. 
Geological Survey, at about 70 different gaging stations, 
showing the mean value of the reduction coefficient to range 
for different stations from 0.78 to 0.95, with an average for 
all stations of 0.85; and an elaborate measurement of the 
Susquehanna at Harrisburg, November 2, 1903, by engineers of 
the same survey, showed a mean coefficient, to be applied to 
velocities observed one foot below the surface of 0.85 for 14 
verticals, with a range in individual verticals from 0.76 to 
0.91 (Eng. News, Jan. 14, 1904.) 


Apparently those conditions which tend to depress the 
locus of maximum velocity in a vertical (Art. 11) will tend 
also to increase the numerical value of the coefficient, which 
should therefore be higher for relatively narrow, deep channels, 
than for wide, shallow ones; and Grunsky has given a graded 
set of mean values for the coefficient, diminishing from 1.03 
f 02 ’ streams whose width is 5 times the average depth, to 0.91 
foi* streams whose width is 30 times the average depth, and to 
0.82 for streams whose width is 100 times the average depth. 

(Engineering Recoi'd, Mar. 7, 1896). The experimentally 
determined coefficient would vary somewhat accordingly as the 
so-called surface velocity was measured close to the surface, 
as with a thin block of wood; or several inches, or perhaps a 
foot below the surface, as with some current meters, and pub¬ 
lished data do not always make clear the exact depth of the 
observation; but as a fair general approximation for river 
cross-sections we may write:- 


Mean velocity in any vertical 

Velocity at or within 1 foot of surface in same vertical 


0,85 








































































































































■ 

■ 



























































24 


20. Velocity Measurements at Probable Denth of Mean Velocity 
past Vertical . Assuming the vertical velocity curve to be an 
ordinary parabola with horizontal axis, the locus of mean velocity 
will theoretically vary from 0.58 dej>th, with maximum velocity at 
surface, to 0.65 depth, with maximum velocity at 3/l0 depth. 

It appears, therefore, that the thread of mean velocity in any 
vertical is likely to be found, in general, in the near vicinity 
.of 6/l0 depth, and this accords with mean results actually ob¬ 
tained from very many different series of experiments under 
widely varying conditions. 

It is important to notice, however, that while 6/l0 depth 
is a good average value for the locus of the mean velocity in 
verticals, yet that for particular cross sections, and still 
more for particular vertical curves of any single cross-section, 
there may be wide departures from that value. Taking the 
average relative depth of mean velocity for all verticals at 
a single gaging station, and comparing these averages for dif¬ 
ferent streams and stations, the range will be well covered by 
0.55 - 0.65; and yet, for the Mississippi at Carrollton, by the 
observations of 1883, the relative depth was 0.66 at low water 
and 0.77 at high water, while it varied from 0.3 to 0.8 in 
individual curves obtained in 1882 (Trans. A.S.C.E., Vol. 34, 
p. 384). 

For the Niagara, by the measurements of 1891-92, the 
relative depth varied in different verticals from 0.47 to 0.68 
in one cross-section, and from 0.50 to 0.64 in another, al¬ 
though the general average was 0.58 for one cross-section and 
0.60 for the other (Eng. News, March 2, 1893). 

At the Cornell flume, in 1900-01, Murphy found the rela¬ 
tive depth in different individual experiments to vary from 
0.50 for a water depth of 6 or 8 inches, to 0.73 for a water 
depth of 8 or 9 feet, although the average of 31 experiments 
was 0.64 (U.S.G.S. Water Supply and Irrigation Paper, No. 95, 
p. 99). 

In an elaborate measurement of the Susquehanna at Harris¬ 
burg by engineers of the U. S. Geological Survey, November 2, 

1903, the relative depth of the thread of mean velocity varied 
among the twenty verticals from 0.51 to 0.72, although the 
average of all was 0.61 (Eng. News, Jan. 14, 1904); and Hoyt 
and Grover give data for over seventy gaging stations of the 
U. S. Geological Survey, covering about 900 vertical velocity 
curves, the mean position, for individual stations, of the 
thread of mean velocity varying from 0.58 depth to 0.73 depth, 
with an average for all stations of 0.61, the error in the mean 
velocity past the vertical, when assumed to be given by the 
observed velocity at 6/l0 depth, not exceeding 6^ for the 
mean of any station, and averaging zero for all the stations 
taken together. 







25 


Von Wagner has given data for nine sets of curves (64 in 
all) for the Danube, Rhine, Elbe, Weser and Oker rivers, rang¬ 
ing from 425 meters down to 14 meters in width, showing an 
average relative depth of the thread of mean velocity for all 
of 0,597, the range between different sets being only from 
0,58 to 0.62, although the locus of maximum velocity varied 
from the surface down to 0.25 depth (Min. Proc# Inst. Civ. 

Engrs•, Vol 71, p. 90). 

It' is evident then that, although a point measurement at 
6/lQ depth is likely to give a close approximation to the mean 
velocity past the vertical for an ordinary natural channel, 
yet, if the best results are sought, some discrimination should 
be used as to varying the relative depth of measurement for 
different channels, and even for different parts of the same 
channel; and it is best that, by special experiments at each 
gaging station, the proper relative depth should be found for 
different verticals and river stages, or, what amounts to the 
same thing, that the proper reduction coefficients to apply to 
observations at 6/l0 depth should be determined. The practice 
of measuring at s/lO depth in large rivers is common with gov¬ 
ernment engineers in this country, a reduction coefficient 
sometimes being employed and sometimes not, according to cir¬ 
cumstances . 

It is well to remember that those conditions (noticed in 
Art. 11) which tend to depress the locus of the thread of maximum 
velocity in the vertical tend also to depress that of mean veloc¬ 
ity, and vice versa; consequently, in deep, narrow channels we 
may expect the thread of mean velocity to be at a relatively 
lower level than in shallow, wide ones. Unusual roughness of 
the bed, making the decrease of velocity towards the bed more 
pronounced than otherwise, also tends to lower the thread of 
mean velocity. As the result of elaborate studies for the 
U. S. Geological Survey (Water Supply and Irrigation Paper, 

No. 95, p. 137). Murphy concluded that for an approximately 
straight, regular channel, with few obstructions, 

(a) "In a broad, shallow stream from 3 to 12 inches in 
depth, and having a sand or fine gravel bed, the 
thread of mean velocity is from 0.50 to 0.55 depth 
below the surface." 

(b) "In broad streams from 1 to 3 feet in depth, and 
having gravelly beds, the thread of mean velocity 

is from 0.55 to 0.60 of the depth below the surface." 

(c) "In ordinary streams where the depth varies from 
about 1 to 6 feet, the thread of mean velocity is 
about 0.6 below the surface." 

For the Merrimack measuring flume, Lowell, width 48 feet, 
water depth about 10 feet, Lynch and Wheeler (M.I.T. 1894, 1895) 
found the mean relative depth to be 0.67 for the thread of 
mean velocity.. 
















































* 

* 














26 


With the maximum velocity below l/3 depth, as occurs in 
ice covered or other closed channels flowing full, and near the 
side of seme open channels, there may be expected two loci in a 
vertical for the thread of mean velocity, and JLn open channels 
for such curves a fair approximation to the mean velocity may 
be obtained by an observation at either 5/8 depth or l/iO depth. 

21. Velocity Measurements at Mid-Depth in Vertical. 

Since the mean velocity in the vertical frequently occurs at a 
relative depth differing considerably from 6/l0, and since for 
close results a reduction coefficient must then be applied to 
velocities observed at 6/l0 depth, and since there has been 
claimed to be greater constancy in the velocity at mid-depth 
than at 6/l0 depth, engineers have frequently preferred to make 
their measurements at the higher level. 

Humphreys and Abbot considered their extensive observations 
in the Mississippi to prove the ratio of mid-depth velocity in a 
vertical to mean velocity in the same vertical to be a "sensibly 
constant quantity for all practical purposes" with a mean value 
of about 0.98, but their conclusions have not been confirmed by 
later observers. Starling states (Trans. A.S.C.E., Vol. 34, 
p. 382), that the measurements of the Mississippi River Commis¬ 
sion in 1882 showed "the ratio between the mean of the mean 
velocities in all the verticals at each observation point and 
the mean of the mid-depth velocities in the same verticals" to 
vary from 0.94 to 0.98, the mean of all being about 0.96. 

To test the constancy of the mid-depth velocity Cunningham 
made special experiments in the Ganges canal, measuring the same 
mid-depth velocity forty-oight times in quick succession with 
double floats and twelve times with current meter, obtaining a 
ratio of mean to raid-depth velocity ranging from about 0.92 to 
about 1.08 (Min. Proc. Inst. Civ. Engrs., Vol. 71, pp. 15, 19). 

Observations below the Vyrnwy reservoir, by Ellis in the 
Connecticut river near Thompsonville, by the Mississippi River 
Commission, as quoted by Starling, by the U. S. Geological 
Survey in the Esopus and six other streams considered in con¬ 
nection with the water supply of New York City, by Wheeler and 
Lynch in the Merrimack canal at Lowell, and others, show mean 
values of this ratio for the various gaging stations ranging 
from 0.92 to 0.98. 

Observations by Allen and Griffin (M.I.T. 1908) as to the 
comparative constancy of velocities at successive tenths of the 
depth in selected verticals in four cross sections (Charles 
River at two points, Sudbury Aqueduct, and Hamilton Flume, 
Lowell) show for three of these greater constancy at mid-depth 
than at any other depth. It seems to the writer that such 
published information as is available indicates some advantage 
in constancy for mid-depth, but that the advantage is not great 
over other depths in the middle third of the vertical, and that 
the constancy is not such as to preclude the necessity in close 











27 


work of using specially determined reduction coefficients. 
For approximate results, however, we may assume:- 


Mean velocity past a vertical line 


Mid-depth velocity past same vertical 


= 0.95 


22, Single-Point Velocity Measurements at Sundry Depths 
in Verticals, Whether 6/lQ depth, mid-depth, or the surface 
be chosen for single-point measurements, it has been shown that 
a reduction coefficient must usually be applied for accurate 
results; consequently, other depths than those may answer 
about equally well, and have occasionally been employed. It 
has already been mentioned (Art, 18) that all meter observa¬ 
tions in the 1897-98 gagings of the Niagara river were made 

at 5/l0 depth (Jour. Wes, Soc. Engrs. 1899, p. 459); and 
Starling states that in some of the Mississippi river gagings 
observations have been taken at various depths, as might be 
most convenient, appropriate reduction factors being used 
(Trane. A.S.C.E., Vol. 34, p. 383). 

23. Velocity Measurements at Two Points in Verticals. 
Assuming the typical vertical velocity curve to be a common 
parabola with horizontal axis, several formulae may be derived 

for expressing the mean velocity past the vertical in terms 
of two measured velocities. Cunningham presented four such 
formulae in his paper on experiments in the Ganges canal, two 
of which are as follows (Min. Proc. Inst. Civ. Engrs., Vol. 71, 
p. 18):- 

(a) Mean V past vertical s V at surface + 3 (V at -§• depth) 

4 


(b) Mean V past vertical - V at 0,211 depth + V at 0.789 depth 

2 


or approximately 


s V at To depth + V at To depth 

2 


The first has the advantage that the velocities are meas¬ 
ured at the highest levels possible for such a formula. With 
the second the computation involved is simple; with floats it 
would be possible to run a connected pair whose common velocity 
should be the mean in the vertical; and with the current meter 
the specified locations for the instrument are definite. 

Prof, Von Wagner applied this formula to a number of measured 
curves for the Weser, Elbe, Rhine, Danube, etc., and found it 
in most cases to agree perfectly, the differences never being 
more than 2.5J? (p. 89 of above reference). Hoyt and Grover 
give the results of applying the formula to a total of 461 
vertical velocity curves determined since 1905 at 33 different 
gaging stations of the U. S. Geological Survey, from which it 
appears that the mean correction needed for any station to make 
the values for mean velocity past the vertical by formula agree 













28 


with the values given by the curves themselves in no case ex¬ 
ceeded 3/b, and for the whole number of stations taken together 
averaged practically zero. (River Discharge, p, 50)• 

Assuming, as a rough approximation, the vertical velocity 
curve a straight line from surface to bed, the mean of the 
nominal top and bottom velocities is sometimes taken as the 
mean past the entire vertical. The approximation is seldom 
warranted, however; observations cannot be made strictly at 
the surface or bed with meters, and the distance from surface 
and from bed will vary with the construction of the meter; the 
immediate vicinity of the bottom is the poorest of all posi¬ 
tions in the vertical for velocity measurements, because of 
irregularities of the bed, eddies and pulsations; and correc¬ 
tion coefficients for this method are neither well known nor 
likely to be constant. The mean of top and bottom measure¬ 
ments should therefore seldom be employed, unless possibly 
for wide, shallow streams having a smooth bed. 

24. Velocity Measuromentg at Three Points in Vertical. 

If velocities be measured at three points in the vertical, say 
at or near the top, middle, and bottom, their values plotted to 
scale, and a smooth curve drawn through the points thus located, 
a rough approximation to the vertical velocity curve will be 
obtained, and with no particular assumption as to its geometri¬ 
cal shape the mean velocity past the vertical can be found by 
determining the area of the curve by planimeter or otherwise 
and dividing the area by the total depth. 

If the curve be regarded as approximately represented by 
a broken line, then, by the trapezoidal rule, letting T, M, B, 
and V represent the three measured velocities and the mean 
velocity, respectively, 

V = T + 2 M + B 
4 

If, however, the curve be assumed to be better represented 
by a parabolic arc passing through the observation points and 
having a horizontal axis, then 

V=T+4M+B 
‘- r —~ 

The same objections hold here to the use of top and bottom 
velocities that were mentioned in Art. 22, and a slight error 
is bound to result from the fact that velocities cannot be 
measured strictly at the surface or bed; but if for the above 
observation points there be substituted 2/l0 depth, 8/l0 depth, 
and the probable depth of the thread of mean velocity (ordina¬ 
rily 6/l0 depth), then, with the above assumption of a para¬ 
bolic arc, we shall have, very closely, 

Mean V 2 s 

past vertical = V at /<=> depth + 2 (V at say /o depth) + Vf at 73 depth 

4 








29 


25. Determination of Mean Velocity past Vertical 1'rcm Curve. 
The most accurate method of determining the mean velocity past 
the vertical with current meter, especially where artificial or 
unusual natural conditions prevent the ordinary distribution of 
velocities, is to observe velocities at enough points in the 
vertical so that when they are plotted to scale the vertical 
velocity curve may be drawn through the points thus determined. 

The area of this curve may be found by planimeter or by any 
other suitable method, and the area divided by the depth gives 
the mean velocity. The scale for velocities should be made 

no larger than appears necessary, and an irregular curve may 
be drawn so as to pass through all the points, or a smoother 
curve may be interpolated, but the former method seems to the 
writer the better. Since observations cannot be taken di¬ 
rectly at the surface or bed, the curve will have to be ex¬ 
tended by judgment beyond the plotted points, but its probable 
direction is usually evident. It may be well to remember, 
though, that the velocity probably diminishes pretty rapidly 
in the immediate vicinity of the bed, perhaps more rapidly 
than the general trend of the curve would indicate. 

It is well to occupy at least five points in the vertical 
in any case, and there is a possible advantage, for the purpose 
of comparing different methods of computation, in including 
2/l0, 5/l0, 6/l0 and 8/l0 depth; frequently, observations 

are made at each tenth of the total depth, and sometimes on 
deep streams, at still shorter intervals (see Art. 17). 

The time required for applying this method throughout a 
stream gaging usually limits its application to channels of 
moderate size, or to certain selected verticals in a large 
stream, the purpose in the latter case being to determine, 
after sufficient repetitions, proper average values for reduc¬ 
tion coefficients. 

26. Integration of Velocities Past Vertical. If the 
current meter be moved at a uniform rate through a vertical, 
the total number of revolutions of the wheel and the total 
time interval being noted, the average number of revolutions 
per second may be found and, from the rating of the meter, the 
corresponding velocity of the water. This will be approxi¬ 
mately the mean velocity past the vertical. There are certain 
minor defects in this method, however, as follows:- 

(a) The cons traction of meters prevents their quite 

reaching the bottom of the vertical, with some types 
within six inches or a foot, or even more, and the 
slow' moving velocities near the bed are not included 
in the result, which is therefore too large. In the 
gaging of the Susquehanna at Harrisburg, November 2, 

1903, by engineers of the U. S. Geological Survey, 
the integration method gave about 2 % higher discharge 
than that by vertical velocity curves. 








30 


(b) The relation between revolutions of the meter and 
velocity of the water is not represented by a right 
line, but by a slightly curved line; consequently, 
the greater the variation of the velocities past the 
vertical, the less accurately the mean speed of the 
meter wheel corresponds to the mean velocity of the 
water. 

(c) The vertical movement of the meter itself affects the 
indicated velocity, decreasing it in the case of the 
Fteley meter, but increasing it in the case of the 
price or any other type of meter adjusted so that it 
is free to head in the direction of the resultant 
velocity. The relative error is greatest in slow 
currents, and to make it negligible the vertical 
speed of the meter must be mad© small. 

(d) In slow currents the meter may, in its vertical motion, 
especially near the bed, encounter velocities so low 
that its indications are unreliable or that it en¬ 
tirely ceases to revolve. 

(e) It is difficult to obtain, without a complicated out¬ 

fit, or at least the aid of an assistant, such regu¬ 
larity of vertical motion as to make negligible the 
error due to variable speed (see description of 
Harlacher*s apparatus for moving a meter with uniform 
velocity: U.S.G.S. Water Supply and Irrigation Paper, 

No. 95, pp. 38, 39). 

Nevertheless, in spite of these defects, the method of 
vertical integration, in the hands of a judicious and prac¬ 
ticed observer, may give a high degree of accuracy, even with¬ 
out any special contrivance for ensuring uniformity of motion. 
Where conditions are poor, as where the channel is crooked, or 
irregular in outline, or the surface is frozen or otherwise 
obstructed, thus preventing the usual relations between veloc¬ 
ities past the vertical; and where time is wanting for com¬ 
plete point measurements in verticals,- under such circumstances, 
if the velocities have not too low a range, the simplicity and 
comparative speed of vertical integration may make it of great 
service. It is also useful as a check upon results obtained 
by other methods. The single operation of lowering the meter 
to the bed may give at once the depth and the approximate mean 
velocity past the vertical. The integration may be made 
either downward or upward, and very commonly the two trips 
are combined in one continuous observation. 

27. Method of Measurement used by Hydrographic Department 
of Hungary. The Hydrographic Department cf Hungary has em¬ 
ployed an ingenious method of determining the mean velocity 
past verticals, called the "detailed" method (Annales des Ponts 
et Chaussees, 1898-3, No. 40). It combines with the speed of 
vertical integration substantially the accuracy cf point measure¬ 
ments, while giving even more complete information than the 










31 


latter as to the variation of velocities past the vertical and 
as to the shape of the curve. The meter is lowered and raised 
at a slow but regular speed by a cable and windlass. As the 
windlass turns it unrolls at a reduced, but proportionate rate, 
a band of paper from a specially arranged chronograph, the 
length unrolled up to any given moment measuring the vertical 
distance through which the meter has sunk or risen in the water. 
The chronograph impresses on the paper band characteristic 
marks at the expiration of each half second of time and of 
each revolution of the meter wheel. 

The procedure resembles that in integration, the meter 
being lowered to the bed and then raised to the surface, thus 
giving two complete determinations for the vertical. While 
regularity of motion is sought, it is far less essential than 
in ordinary integration. Intervals of two centimeters height 
on the paper, ordinarily corresponding to twenty centimeters 
of actual vei'tical distance in the water, are marked off, and 
opposite each such interval the chronograph record enables one 
to compute the meter speed in revolutions per second, from 
which in turn the corresponding velocity of the water may be 
found from: the rating of the meter. The data are then known 
for plotting the vertical velocity curve, which can be traced 
with greater or less refinement by varying the lengths of the 
intervals into which the record is divided. The extremes of 

the curve at surface and bed must be drawn by prolongation as 
in ordinary point measurements. While traversing a vertical 
in 12 or 14 meters depth of water would require nearly or 
quite an hour by point measurement, it is accomplished in 
about five minutes by the "detailed” method. 

28. Measurement of Mean Velocity past Vertical by Rod or 
Tube Floats. By.the use of a wooden rod or closed metal tube 
(usually of small diameter) so weighted at the bottom as to 
float with axis vertical and but little exposure above the 
water surface, and reaching as near the bed of the channel as 
circumstances will permit, a close approximation may be made 
to the mean velocity from surface to bed along the path trav¬ 
ersed by the float, if the depth be uniform, as in an artifi¬ 
cial channel. The method is frequently applied also to 
natural channels, but with less satisfactory results. 

Inasmuch as the float cannot be allowed to reach quite to 
the channel bed, it is not acted upon by some of the slow moving 
water near the bed, and therefore tends to move faster than it 
would do if more deeply submerged. But it does not strictly 
move with the mean velocity of the water, even for the depth of 
actual immersion, for the following reason: The float is sub¬ 
ject to pressures proportional to the square of the relative 
velocities of the water at different points with respect to it, 
beyond a certain speed these pressures presumably being exerted 
simultaneously against the up-stream side of the float in the 
upper part of its length and against the down-stream side in 
the lower part of its length; and the float will not reach its 


4 





32 


full normal speed until equilibrium has been reached between 
these different pressures. It is evident that this equilib- 
rium may exist , however, when the speed of the float is some¬ 
what different from the mean velocity of the water for the 
depth of immersion, the latter velocity being simply the 
arithmetical mean of all the different velocities from the 
water surface to the bottom of the tube. 

As the result of mathematical investigation, making the 
ordinary assumption of the vertical velocity curve being a 
parabola, Cunningham concluded that the float velocLty would 
always be somewhat less than the mean velocity of the water 
for the depth of immersion, and that to measure the mean 
velocity past a vertical the float should be immersed only 
about 0.94 of the wat8r depth (Min. Proc. Inst. Civ. Engrs., 
Vol. 71, p. 22). 


Elaborate experiments by Francis at Lowell in the Tr©mont 
measuring flume (widths in different series of experiments 
about 27 and 13 feet,respectively, and water depth 9 feet, 
more or less) showed, however, that the mean velocity of a 
3eries of floats for a complete gaging nearly always exceeded 
the mean velocity of the water past the entire cross-sectibn 
as found by weir measurement, and led to the following cor¬ 
rection coefficient:- 


If V = true mean velocLty past vertical from surface to bed, 
V 0 = observed velocity of tube, 
d ~ mean water depth along path of tube. 


d f * depth of immersion of tube, 

then C = L- 58 1 *“ 0.116 / f~d-d» - 0.1] 

Vo W d / 

It will be noticed that the value of this coefficient 
rises to unity only when the depth of float Immersion reaches 
about 0.99 of the water depth. Probably this formula would 
not strictly hold for other than rectangular channels, nor in 
any case where there wa3 an abnormal distribution of velocities 
past the vertical; nevertheless it has been widely applied to 
various conditions in practice. 

d-d 1 

In Francis’ experiments the ratio ~~ did not exceed 0.12, 

d 

and to apply his correction formula to cases in which the rela¬ 
tive clearance below the float much exceeds 0.12 of the depth 
involves an error that is not well known. On repeatedly 
running at short intervals a series of floats of different 
lengths, from 7 to 11 feet, over the same course, in water 
about 12 feet deep, Brown found that, assuming Francis’ co¬ 
efficient correct for 5 % clearance below the float, 















, the numerical value of the coefficient then 


33 


/ d-d’ 


0.05 


being 0.986, the successive values best representing the re¬ 
sults of his own experiments were as follows (Wis. Engr., Vol.6):- 

for 10^ clearance C = 0.969 

" 20/b " C = 0.942 

" 30X " C = 0.919 

" 35^ " C = 0.908 

Using tube floats of different lengths, which were run 
alternately with standard long tubes submerged about 95^ of 
the water depth, Dort and Faulkner, M.I.T. *09, made a large 
number of observations similar to those by Brown. Their 
runs were made in two canals at Lowell,- the Merrimack, about 
48 feet wide, with 10 or 11 feet depth of water, observed 
velocities ranging between 2 and 3 feet per second; and the 
Boott, about 41 feet wide, with 8 or 9 feet depth of water, 
velocities ranging mainly between 4 and 5 feet per second. 

For clearances less than 35/o their mean coefficients for the 
Boott canal agreed best with those by Francis 1 formula, in no 
case differing from the latter by more than but for the 
Merrimack they agreed much better with Brown’s values than 
with Francis’ , being practically identical with the former 
throughout. 


29. Proper Spacing of Verticals. The choice of hori¬ 
zontal intervals at which observations shall be made is more 
or less arbitrary, but should have general reference to the 
accuracy to be aimed at in the result, the steadiness of the 
stream, and the time likely to be required for.a complete 
gaging. The judgment may be aided by having in mind the 
transverse velocity curve (Art. 32), since such spacing as would 
determine this satisfactorily should be otherwise acceptable. 
This principle would lead to observations being taken at con¬ 
spicuous points of the transverse profile of the stream bed, 
relatively closely near the banks, and farther apart where the 
velocity changes slowly and regularly. Computation of dis¬ 
charge without plotting is facilitated by spacing velocity 
stations as well as soundings at regular intervals, and the 
general instructions of the U. S. Geological Survey (Water 
Supply and Irrigation Paper No. 94, page 19) call for measure¬ 
ment of velocities in each vertical in which a sounding is 
taken, except where the change in velocity is small, and there 
in alternate verticals. This would lead to intervals ranging 
from 1 foot to say 50 feet, according to the size of stream 
(Art. 5). 


The practice of the Hydrographic Department of Hungary is 
not to take intervals greater than from 20 to 40 meters, even 
on a stream as large as the Danube. On the Connecticut, where 
from 1,000 to 1,500 feet wide, Ellis used intervals of about 
100 feet. On the Niagara, width about 1,800 feet, Haskell 







had raster stations about 80 feet apart. About 1890 the prac¬ 
tice on the Mississippi river was to space meter stations 300 
feet apart, but the interval was afterward reduced to 150 feet 
comparative computations of discharge were made by Captain 
Townsend in 1893, using the observations at 300-foot and 
150-foot intervals, respectively, with a resulting difference 
ordinarily less than 1/C and never exceeding about ?>%. 

30. "Flanking" Method of Measuring Velocities. As a 
check upon measurements otherwise obtained, a method known as 
"flanking" has been used in the Mississippi river, resembling 
in a rough way, in principle, the "detailed" method described 
in Art. 27, but involving a horizontal instead of a vertical 
motion of the meter. The meter is submerged 15 or 20 feet, 
so as to be below disturbance from the launch, which is then 
worked slowly across the river, being kept as closely as pos¬ 
sible on the established transverse range, head up-stream, 
until the opposite bank is reached, when a return trip is 
made, the two trips constituting one complete observation. 

At regular distances, fixed in advance by means of shore 
ranges, readings are taken of elapsed time and revolutions 
of the meter, from which may be determined the average re¬ 
sultant velocity of the water in each section of the path 
of the meter. The observed velocLty may be corrected for 
the lateral motion of the boat as follows: 


If V 1 = velocity given by meter, 

V" = velocity due to lateral motion of boat, 

V = absolute velocity of water, 
a ~~ 2 ~ 


then V = 


M 


V' 


- V" 


The mean of the results obtained in opposite trips is adopted 




31. Continuous Integration Over Entire Cross-Section. 
Where one may operate the current met far from a solid support, 
such as a beam spanning the channel, or by wading, it is often 
practicable to integrate velocities over substant.tally all 
parts of the cross-section in one operation, and thus to de¬ 
termine at once its mean velocity, and excellent results are 
often obtained in this way in a very brief time. The wetted 
cross- 3 ection should be conceived as subdivided by imaginary 
lines into vertical or horizontal strips of equal width, and 
the motion of the meter through these so directed that, as 
nearly as possible, equal ele/nentary areas shall be centrally 
traversed in equal times. If the strips are vertical, the 
motion is often a diagonal one, the meter being started at the 
surface at one side of the channel and moved obliquely down at 
a slow, uniform rate to the bed, returning thence obliquely to 
the surface, thus covering one strip, and so on through the 
remaining strips to the farther side of the channel. 








35 


On the lines of the aqueducts supplying the Boston metro¬ 
politan district with water are permanent gaging stations at 
which frequent current meter measurements are made by a con¬ 
tinuous integration of consecutive 6-inch horizontal strips. 

The path of the meter through these successive strips is me¬ 
chanically controlled (see Eng. News June 12, *02, p. 488, 

Fig. 3), and the speed of motion is determined by the operator, 
who is aided by a metronome beating seconds and is checked at 
intervals by an assistant with a stop watch. The downward 
and return upward integrations usually vary somewhat in the 
results, but the average error of their mean is believed not 
to exceed 0.25^. If the water level is such as to give a 
surface strip of materially different width from the standard, 
each lower strip is integrated separately, the respective 
mean velocities of the strips are plotted to scale, and the 
extension of the resulting curve gives the probable mean 
velocity of the surface strip. 

32. Transverse Velocity Curve. If the mean velocity 
past each of a series "of* vertical lines, regularly distributed 
across the section, be plotted to scale, in its proper posi¬ 
tion as an ordinate from a horizontal line representing the 
water surface, there will be determined a curve which for 
convenience will be called a "transverse velocity curve", and 
which is often employed in the computation of discharge. In 
general the ordinates to this curve will be greatest in the 
deepest part of the channel, diminishing as shoal water is 
approached, or the banks are neared, and very rapidly close 

to the banks. 

33. Conception of Discharge as Represented by a Solid. 

If a differential area in the cross section be imagined as the 
base, and the velocity past that area the altitude,,the solid 
constructed with these two dimensions will represent a differ¬ 
ential portion of the discharge. If the entire cross-section 
be similarly treated, the resulting solid will represent the 
entire discharge. The method which determines the volume of 
this solid with the closest approximation, determines the dis¬ 
charge most accurately. The computation of the discharge can 
often be made without the aid of any plot, as where soundings 
and their velocities are measured at regular intervals across 
the stream; but even then it is an advantage, as a check 
against gross errors, and in other cases it is a necessity, to 
plot the cross-section, if otherwise than rectangular, the 
transverse velocity curve, and sometimes a transverse discharge 
curve. 


34. Subdivision of Solid i nto S lices. The solid repre¬ 
senting the discharge is conceived to be divided into slices, 
usually of equal width, by parallel planes cutting it in one 
of three directions;- 

(a) Parallel to plane of water surface; i.e, horizontal. 

(b) Parallel to plane of cross-section; i.e. vertical, and 

transverse to stream. 






















































































































36 


(c) Perpendicular both to plane of cross-section and to 

that of water surface; i.e. vertical, and 
longitudinal with stream* 

Usually the slices are taken of definite width, varying 
say from one foot upward according to circumstances, and some 
suitable formula is applied to finding their approximate 
volume. The curved surfaces are not made up of straight-line 
elements, and the prismoidal formula is not strictly applicable. 
Frequently the volume of a slice is taken as the area of a mid¬ 
section multix>lied by the thickness of the slice. The thinner 
the slices the more accurately may the computation be made, 
and in some methods they are even treated as of only differ¬ 
ential thickness. No more refinement is really necessary, 
however, than shall be consistent with the completeness and 
probable accuracy of observations. 

Case (a) Each slice, except the bottom one, in natural 

channels, will have three plane faces, with 
sides and down-stream end curved. The top and 
bottom face will each be limited by a horizon¬ 
tal velocity curve; but velocities are not 
U 8 uall 3 r measured at such depths as directly to 
define those curves, and this method is not 
therefore commonly used. 

Case (b) The slices will in general have three plane 

faces, with sides and bottom curved. The two 
principal faces of each slice, foi’med by the 
parallel cutting planes, will be bounded by 
contours of equal velocity, which can be traced 
on the cross-section in the case of numerous 
and well distributed point measurements only. 
Their areas can readily be measured by plani- 
meter. The residual volume at the apex of 
the solid is likely to be of less height than 
the slices, and its contents can be only rough¬ 
ly computed. This method is described by 
Unwin (Hydraulics, p. 289), but is probably 
not often used. 

Case (c) Except next the sides of the channel each slice 
will have plane faces for its base (considered 
as a part of the cross-section), top and two 
sides, with a curved surface for the remainder 
of its exterior, this curved surface not being 
of any definite geometrical shape. It is 
this arrangement of slices upon which practical 
computations of discharge are most often based, 
although various procedures are adopted for the 
details of the work. 

35 , Harlacher f s Graphical Method of Computing Discharge. 
Referring to Fig. 8 it will be seen that, if D is the depth in 
a particular vertical, and V is the mean velocity past that 






37 


vertical, then the discharge 
past a strip of the cross- 
section dx in width will he 
V D dx. If, therefore, one 
were to lay off a series of 
ordinates from the line repre¬ 
senting the water surface, 
making each of a length equal 
to its respective V D, and 
extending the operation from 
shore to shore, the area of 
the curve thus formed would 
evidently represent the total 
discharge. 




Harlacher has shown a graphical method of constructing 
this curve (’’Die Messungen in der Elbe und Donau, und die 
hydroraetrischen Apparate und Methoden des Verfassers” , Leipsig, 
1881; see also "Hydr aulique ” , par A. F lam ant, Paris, 1891, 
p. 359) the determination of the discharge being thereby re¬ 
duced to the planimetric measurement of an area (Fig. 9):- 
If, as before, V represents the mean velocity past any vertical 
D, then, as before, V D dx represents a differential portion 
of the discharge, and the entire discharge Q, =J*V D dx, the 
integration extending from one bank to the other. Lay off a 
convenient arbitrary distance K, make E C - V, and draw a line 
from C parallel to A B, cutting E B produced at P. By con¬ 
struction V D = (E P) K; conse^uentlyJ'V D dx = Kj*(E P) dx = K 
multiplied by the 
area of the discharge 
curve which is the 
locus of all points 
P determined as above. 

If the area of 
the discharge curve, 
be measured by plan- 
imeter in square 
inches, and the 
length of K in inches, 
velocities being plot¬ 
ted to a scale of S 
feet to one inch, 
horizontal distances 
to a scale of S’ feet 
to one inch, and 
depths to a scale of 























38 


S M feet to one inch, then the measured area must be multiplied 
by K S S' S" to give the discharge in cubic feet per second. 

36. Procedure in Computing Discharge for Irregular^ Dis ¬ 
tributed Verticals, as in Rod or Tube Float Measurements . The 
current meter can be very exactly 

placed in selected verticals, ordinarily spaced at equal inter¬ 
vals across the channel, and the drawing of a transverse velocity 
curve is not always then necessary for computing the discharge, 
although it is always useful as a check against error. Floats, 
however, seldom pass the upper and lower cross ranges, which de¬ 
fine their run, at the same distance out from the reference 
line on the shore, and the mean positions of successive floats 
are likely to be separated by more or less irregular intervals; 
moreover, closely adjacent runs often give materially different 
velocities. Therefore, and especially for narrow channels, in 
which the floats are apt to be run at rather short lateral in¬ 
tervals, the drawing of a transverse velocity curve is often 
a practical necessity. 

The mean position of a float must usually be taken as 
fixed by the mean of its distances out at the crossing of the 
upper and lower ranges. Whatever its path between those ranges, 
whether normal to them, oblique, or curved, its average veloc¬ 
ity normal to the ranges,- and it is only that component which 
is desired for computing discharge - is given by dividing the 
norma], distance between the ranges by the elapsed time in mak¬ 
ing the passage. This velocity may then be reduced to mean 
velocity in the vertical by applying the proper correction 
coefficient. 

If, now, the values obtained for mean velocity in the 
various verticals be plotted to scale, in their relative posi¬ 
tions, as ordinates from the straight line representing the 
transverse profile of the water surface at its mean position 
during the gaging, points will thus be fixed for drawing the 
transverse velocity curve. A reasonable curve can seldom be 
drawn, however, which shall pass through all the plotted 
points, since closely succeeding ordinates may vary materially 
in length on account of irregularities of current, in which 
case a mean curve must be interpolated. This may first be 
drawn roughly by eye, following the general trend of the 
points, and then adjusted and redrawn through the successive 
groups into which the points either naturally fall or may 
arbitrarily be assigned by arranging that in each group the 
plus and minus ordinates with respect to the curve shall bal¬ 
ance (Fig. 10). The plus values in a group may be set off 
successively on the edge of a strip of paper, and similarly the 
minus values, and the balancing is thus easily tested. 

' The smaller the number of points taken in a group, the 
less smooth the curve will be, but if used with discretion, 
as an aid to the judgment rather than as a substitute for it, 
this method will result in a curve satisfying the eye and 










39 



giving practically the same computed discharge as any other 
curve passed through the same set of points somewhat differ¬ 
ently grouped. The extreme ends of the curve must he sketched 
in by prelongation, and it should be remembered that the ve¬ 
locity usually falls away rather rapidly in the immediate 
vicinity of the banks. 

Having thus obtained a transverse velocity curve, uni¬ 
formly-spaced verticals may then be chosen as desired, their 
velocities scaled off, and the computation of discharge pur¬ 
sued as in the following article; or, if preferred, the graph¬ 
ical method of Harlacher (Art. 35) may be employed. 

37. Procedure in Computing Discharge for 
Uniform^ Distributed Verticals . Harlacher * s . 
method is theoretically exact, but practically 
the construction of his discharge curve is de¬ 
pendent upon the number of verticals in which 
depth and velocity have been measured, and is 
subject to inaccuracies of plotting. Moreover, 
a graphical method is impracticable for use in 
the field, where immediate determination of the 
discharge may be desired. Consequently it is 
well to notice modes of computation for which 
plotting is not essential. 

Conceive the entire cross-section subdivided 
into vertical strips of moderate width . Let 
Fig. 11 show a single stripoT~wTdtnT b , and 





































40 


suppose the "bed level , thus giving a rectangle , of depth D. 
Past an elementary portion of this strip, of width dx, the 
discharge is V D dx, and past the entire strip it is 


/* A V D dx, 

Jo r 

- »x 


V dx 


= D times area of corresponding portion of transverse 
velocity curve, 

= b D times mean ordinate to said curve, 

= area of strip times mean ordinate to said curve. 

If the bed is not level , and the strip therefore not rectangular , 
the above relation does not strictly hold; but by properly lim- 
iting the width of the strips the error in the computed dis¬ 
charge can be kept small and a satisfactory approximation ob¬ 
tained. 


The following detailed methods of computation, applicable 
to cross-sections of any shape , are now to be noticed:- 


(a) (Pig. 12). Depths D, D f , etc., and mean 
velocities V, V’, etc., known for succes¬ 
sive verticals equally spaced at the com¬ 
mon interval b. 

If we assume D and V, D* and V’,etc., 
to represent approximately the mean 
depths and mean velocities for the strips, 
of common width b, of which the verticals 
are tne center lines, then we may write 

A Q, = b D V = discharge past strip. 

Q = b £ D V = discharge past en¬ 
tire cross-section, 
possibly excepting 
partial strips next 
the banks. 



(b) (Pig. 13). Depths D, D*, etc., 
and mean velocities V, V* , etc., 
known as before in successive 
verticals, equally spaced . 

Taking a strip whose center 
line is D*, and assuming V* to 
be the mean velocity past this 
strip, we have 


A Q a* V 9 times area of strip. 


If, now, e f and f g be treated 
as straight, 


4 Q 


v . b . 



e 

























41 


(c) (Fig* 13}« Taking the double st rip included between 
I) and D* 4 f and treating e~T' g“an3"Ti i j as parabolic 
arcs p 


a q 


2 b + 4B f -i- D a \ | V 4 4 V' + V”\ 
discharge past the double strip. 


(d} (Fig. 14)* Supposing the mean Telocity 
past the vertical center line of each 
strip to have been observed dr scaled, 
then for any strip, as that between D 
and D* , assuming e f straight, we have, 

a <z *> v b 2_± JL1 . 

Evidently this method may be applied 
tc successive single strips, whether 
or not of equal width. 



F/g. /4 



(e) (Figc 15}. Supposing the velocity tc 
have been measured at sufficient points 
in the vertical center line of a strip 
tc permit laying off ordinates at 
proper depths and drawing the v ertic al 
velocity curve , as indicated by cross® 
hatching, then, treating % f as 
straight, 

^ Q s b times area of vertical 
velocity curve. 

It will be seen that, if soundings and mean ve¬ 
locities past verticals have been measured at 
suitable equal intervals in the cross-section, 
methods (at to TcTTlnclusive are applicable to 
computing the discharge without the necessity 
of any plotting. 



F/g. /S 


In any case f if there be an odd strip at the side of the 
cross-section, not covered in the above computations, tne dis¬ 
charge past that strip must be found separately, say by method 
(d) , and added to the amount obtained from the other strips.* 


38. special Methods of Computation Allowable for Rectangular 
Cross-Sections. If the cross —sect ion be rectangular, as is 
oftelfrTrue^of^the head-race or tail-race of a mill or power 
station- then. 





























42 


(a) If vertical strips of equal width be taken* they will 
also be of equal area ? and their mean velocities en~ 
titled to equal weight in averaging* As shown in the 
eexly part of Art* 37, for tne discharge past any 
strip we may write* 

4Q, * D times area of corresponding portion of trans* 
verse velocity curve, 

Q = B times area of entire transverse velocity curve, 
which area may be found by planimeter or other¬ 
wise * 

(h) The methods of Art* 37 are evidently applicable to 
the special case of rectangular sections* 

(c) If A represent the area of the rectangular cross- 
section* and the mean, velocities Y f V 9 V" t etc., 
have been measured past each of n verticals, numerous 
and pretty evenly distributed, then* closely 


39 a Discharge or "Station Hating** Curv es. In the case 
of natural streams'a single gaging usually has but very limited 
value, and where projects of importance are probable for uti¬ 
lizing the water of the stream, as for power or irrigation, 
complete knowledge is desired of the variations in daily flow 
through the year for a series of years* If is neither practicable, 
nor necessary to make daily instrumental gagings for a long 
period of time, and resort is had to finding the relation be¬ 
tween discharge per second and gage height at certain chosen 
gaging stations. Instrumental gagings are there made in at least 
half s dozen different stages, and often in many more, well 
distributed over the total range of fluctuation, and by plot® 
ting gage heights as ordinates and corresponding discharge as 
abscissas points are established through which a smooth curve 
called a discharge curve, or "station rating" curve, may be 
drawn® The assumption then is, that for any gage height within 
•the range of tne diagram, the accompanying discharge in cubic 
feet per second may be read off with satisfactory accuracy. 

Fig. 16 shows a discharge curve for the Susquehanna river 
at McCalls Ferry (Eng. News, Aug. 4, 1904), based upon 38 
gagings. Such a curve having been determined, daily readings 
of gage height may easily be made by an unskilled observer 
living near by* the readings sent to the engineer and by him 
translated into values of discharge. 


Approximately, the physical laws which hold for the 
discharge at any station are expressed by the formulae, 












bage Heigh")" a+ Gage 2 *» El. of W.S. above Mean Sea Level. 


43 


I38'r 


Discha'-qe 


Second -Fee+. 


d Measurements at York Furnace 
a ” " Duncan’s Run Boat Station 

Measurements at Cable Station. 

© Surface Velocities, Small Price Meter 
q ” ” Large ” •* 

« Multiple Faints ” ” * 

Equation of Curve below 6.H. ISO'(y-Hi)*’* (1.032+.05Hy)X 
Appivximcrfe Equation of whole Curve (y=Ul) 2 =2.0SX 
X in 1000 s of sec. tt. 


o © 

8 8 

© cr 

& ?a 


o 



8 

o 

d 

(U 


8 

O 

I 


o 

m 


§ 8 

O O 

O 

»£> h~ 


£ E: £ § g 


o 

cS 


o 

o 

o 

o 

cu 


/v^?. /6 


V - G \j K S 

^ = A C {r s", in which 

Q, s total discharge at a given time past the crees-soctlor,, 
in cubic feet per second, 

A as corresponding area of wetted cross-section, in square 
feet, 

V ss corresponding mean, velocity past the cross-section e 
in feet per second, 

R ss ratio of area of wetted cross-section to its wetted 
perimeter * not including the water line exposed to 
the air. This ratio partly expresses the varying 
effect upon mean velocity past a given area of cross® 
section of merely varying the shape of the section, 
thereby varying the length of the perimeter and its 
relative frictional influence. The value of this 
ratio is called the ^hydraulic radius’ 5 or ^hydraulic 
mean depth”© 

S = inclination of water surface past section in question, 
as expressed by ratio of vertical descent in a given 
distance to the distance itself, the latter measured 
along the water surface. 

C s a variable coefficient, whose value is dependent 

mainly upon the roughness of the channel lining, but 
secondari!y upon the value of and tc a minor extent 






















































































upon other conditions. Other things equal, an in- 
crease in R tends to increase C, while an increased 
roughness of lining tends to decrease C. This co¬ 
efficient may range in value under different circum¬ 
stances from 40 to 150, and even outside those limits 
in extreme cases* 

In using a station rating curve the assumption is ordinarily 
made, and under suitable conditions is justified, that at a 
given station at all times when the water surface is at a 
certain height on tne gage the discharge is the-same,* the 
wetted cross-section, wetted perimeter, surface slope, rough¬ 
ness of lining, mean velocity and, hence, the volume flowing, 
being presumably unchanged. It is very important to notice, 
however, that this is not always true* 

If the bed or banks are unstable,® subject to scour or 
fill, A may change so that at two different dates when Q is the 
same the water level and, consequently, the gage reading shall 
b© materially different. This occurs on the Mississippi and 
other western rivers (Trans* Amer. Soc. Civ. Engrs., Vol. 34, 
p« 389), and in some of the streams of the Alps having unstable 
beds Tavernier notes that it is not rare for the discharge to 
vary in a relatively short time in the ratio of 1 to 2, or 
even more, for the same gage height (Annales des Fonts et 
Chaussees, 190? - IV). 

Again, though at two different date© when the gage height 
is the same A may be nominally unchanged, yet a growth of weeds 
in summer on the bed or banks may have altered greatly the 
value of C, and correspondingly of Q. 

Further, if the station be within the influence of vary¬ 
ing back water due to changing conditions down stream, such 
as the artificial regulation of level in a mill-pond by the 
use of flash-boards, or variations in pond level, due to the 
alternate drawing down and filling up of the pond during the 
twenty-four hours, or ice or log jams, or rank growth of vege¬ 
tation in the channel,® it is plain that A, R and S may be so 
changed that even for the sain© Q the gage height shall vary 
at different times. 

Also it has been well established that water received in 
sudden and heavy freshets from the upper tributaries, of a large 
river comes down the main channel in a “flood wave% at the 
front of which the slop© S is abnormally large, while at the 
rear it is abnormally small. Consequently, at a given station 
on a stream, though the gage height, A, R and C, may all be 
substantially the same at a certain time when the level is 
being rapidly raised by an advancing flood and at a later time 
when the flood i© subsiding, yet Q, may in the first case greatly 
exceed its value in the second. This is clearly shown in the 
discharge curve of the river Fiaza (Fig* 1?). In one or two 
exceptional cases Ellis found the reverse of this law to hold 
true at a station on the Connecticut river (House Ex. Doc. 

No. 101, 46th Congress, 2d session). 







45 



It is clear, then, that much care and caution need to be 
exercised in establishing a permanent gaging station, and in 
interpreting the observations for gage height as measures of 
discharge. 

The normal shape of the discharge curve is considered to 
be a parabola, convex to the axis of gage heights, and tangent 
to that axis at the level of zero flow, which may or may not 
be that of the lowest point in the cross-section. For high 
stages the curve usually becomes approximately straight. 

While a half dozen points may, under favorable circum¬ 
stances, suffice to determine this curve satisfactorily, more 
are needed for low stages than for high, because of the sharper 
curvature as the tangent point is approached. 

In plotting the curve, scales should be chosen that will 
give good intersections of the curve with lines parallel to 
the axes, and that will permit of plotting and of reading off 
values with due precision. 

Since it is quite unlikely that a smooth curve can be 
passed.through all the plotted points, it must be averaged 
among them as well as possible. This is sometimes done by 
assuming a general equation such as Q, ® A + Bx + Cx^ (x rep¬ 
resenting gage height), and finding by the method of least 
squares the most probable values for A, B and C for the given 








































/ 




4 






















. ~ 













V 



























































46 


observations, all these having equal weight* More often, and 
sufficiently well in general, the curve is sketched in by eye 
points being weighted according to any special knowledge that 
may be had as to their probable accuracy. According to the 
practice of the U. S. C-eological Survey, a point obtained from 
any single gaging that varies from a well defined curve given 
by other gagings by more than from 5 to (accordingly as 
the conditions at the station are favorable to good results or 
not) is considered either to be in error or to have been af« 
fected by a change in channel (Water Supply and Irrigation 
Paper, No. 94, p. 22). 

40. Discharge Tables . A discharge curve having been 
drawn, it is usually thought preferable to scale off values 
and put them in tabulated form for subsequent use, rather than 
refer repeatedly to the diagram. The discharges corresponding 
to successive heights on the gage at intervals of, say, tenths 
of a foot are therefore to be read off from the curve, with 
such refinement only as seems warranted by the general accuracy 
of the gagings; and in order to make the table consistent in 
itself the scaled values are corrected, if necessary, by taking 
first, and sometimes second, differences and adjusting them so 
that they either remain constant, or increase with increasing 
gage height, with fair regularity, as seen in the following 
illustration:- 


Gage Height in Ft. 
4.90 
4.95 
5.00 
5.05 
5.10 


Discharge in C.F. 
1390 
1450 
1510 
1575 
1640 


.S. 1st Difference 

60 

60 

65 

6b 


41. Extension of Discharge Curve beyond Limits of Obser ¬ 
vations . In so far as points have been fixed by instrumental 
gagings, the discharge curve can be drawn with confidence; but 
it is frequently desirable to extend it to high or low stages 
in advance of opportunity for actual gagings at such stages. 
Such extensionsneed to be made with much caution, or serious 
errors may result in the estimated discharge. The following 
methods are available 


(a) Extend curve by eye beyond limits of observations. 
This will give a rough approximation, but is not the 
most reliable method. In the lower portion the curva¬ 
ture changes rapidly, and a safer plan for extending the 
curve downward is to assume the lower portion a parabola 
and plot it by coordinates laid off to logarithmic scale 
instead of to natural scale. This will change the curve 











47 


*' 

to a straight line, which may then be prolonged downward 
with considerable accuracy. 

The upper portion of the discharge curve is apt to 
be approximately .straight, but unless its direction is 
well determined by known points its extension may lead 
to wide error. 

(b) The second method takes account of the relation that 
at all stages Q = A V. The values of A can be found with 
any desired accuracy for all stages of flow from an area 
curve based upon soundings and cross-sections of the 
banks. The curve for V usually has but slight curvature 
(normally concave to the axis of gage heights) and can be 
extended upward with more accuracy than the discharge 
curve can be independently extended. This done, the 
products of corresponding values of A and V, scaled from 
their respective curves, will give additional values for 
fixing the extension of the discharge curve. This method 
is unsafe, however, for an extension of the curve down¬ 
ward into low stages, since there the curvature of the 
curve of velocity changes more rapidly than at high stages, 
even reversing under some conditions, and the proper ex¬ 
tension of the curve is uncertain. 

(c) An extension of the discharge curve upward from medium 
to high stages may be based upon the assumption that with in 
that range of heights C and S in the formula Q = A C \l R S 
remain approximately constant. We may then write Q = K A\Z~~R", 

or, since d, the mean water depth, [-- ar9a -J which 

* y \surface width/* 

is easily determined, varies but slightly in a natural 
channel from R, we may write Q, «= K A \Td", for which equation 
the graph will be a straight line and may readily be ex¬ 
tended (see Fig. 18, taken from article by J. C. Stevens 
in Eng. News, July 18, 1907). While this method has 
given good results in particular cases, it must be ap¬ 
plied with caution, since C and S are not strictly constant, 
and may even vary greatly between different stages, es¬ 
pecially between low and high water. 

42. Area, Mean Velocity and Discharge uurves: Theoretical 
Considerations ♦ Reason has been shown why gaging station 
curves are sometimes required separately for area of cross- 
section, mean velocity past the cross-section, and discharge. 

It is true that the drawing of these consists mainly in the 
simple sketching of smooth curves through plotted points. 

These points are bound to be more or less in error, however, 
so that no smooth curve perfectly joins them, and the observa¬ 
tions upon which they are based are time-consuming and expen¬ 
sive. Now the curves possess certain general properties, a 
knowledge of which may often serve to limit the number of 
observations, or to assist in interpreting them or in testing 















48 





their accuracy, it is worth while, therefore, to examine 
"briefly the underlying theory, which is set forth in more 
detail by Tavernier in Annales des Ponts et Chaussees, 

1907 - IV. 

Area Curve. For every computation of discharge based 
upon an instrumental gaging the corresponding area of cross- 
section must be known, either directly from soundings made at 
the time of the gaging, or from soundings made at some other 
time and corrected for the difference in water level. There 
are, then, likely to be at least as many instrumental deter¬ 
minations of area as of discharge, and preferably the profile 
of bed and banks should be closely determined up to the level 
of extreme high water. The area can then be computed for any 
stage, and by plotting a suitable number of points an area 
curve can be drawn, as already noticed, showing the relation 
between gage height and area. Such a curve tends to reveal 
gross errors in measurement or computation of the individual 
areas upon which it is based; serves as a standard to which 
subsequent determinations of area may be referred as a test 
of their probable accuracy, or for revealing the occurrence 
of scour or fill in the channel; and is of aid in drawing, 
and especially in prolonging, the discharge curve. 



































































































































































































49 ! 



Let Fig. 19 represent a stream cross-3ection, and Fig. 20 
the corresponding area curve. 


Let A = area of wetted cross-section at any stage, 

W = corresponding surface width of stream, 

D = corresponding water depth above lowest point of 
bed (equals also gage height when zero of gage 
corresponds to lowest point of bed). 


Considering A = 0 (D) , from any differential strip of thick¬ 


ness d D and area d A it is seen that W = 


d A 
cHE * 


Hence, for 


any position of water surface, the surface width is equal to the 
first derivative of the area with respect to the depth, that is 
to say, to the fraction which expresses the corresponding in¬ 
clination of the curve A =* 0 (D) to the axis of gage heights. 
Thus in Fig. 20, at any point P the surface width W is equal 
to the inclination of the tangent T T* to the Y axis, the co¬ 
ordinate D representing the water depth (gage heignt) and A 
the corresponding area. For a rectangular cross-sect ion, 
in which case the banks will be vertical, W will be constant, 

and the curve A « 0 (D) will be a straight line. In rare cases 

the banks may overhang; but in general they will be inclined 

away from the vertical, with a more or less gentle slope, W 

will increase continuously with the depth, and the curve 
A » 0(D) will be convex toward the axis of gage heights. 

It will be tangent to that axis at zero depth, unless the 
cross-section have a level bottom, in which case the curve 
will meet the axis at an inclination equal to W. 

























50 


The actual inclination on the paper plot of an area curve 
at any point, will of course vary with the relation between 
tne scales adopted for depths and areas, respectively. There¬ 
fore it is necessary to notice that in practically applying 

the relation W = finding the slope of a tangent at any 

point, as P (Pig. 20), consistent scales must be used. If 

be actually one foot, in space, then A A will contain the 
same number of square feet that W does of linear feet; hence, 
if ^ D be laid off as one foot to the scale of depths, A A 
must be laid off equal to W units to the scale of areas. 


Assuming the area curve approximately parabolic (for 
straignt slopes it would be strictly so), we may write, 

A — X D , K being a constant, and x an exponent varying 
with the slope of the banks, 
x = 1 for vertical banks, 

x lies between 1 and 2 for banks concave upward, 
x = 2 for straight, sloping banks, 
x y 2 for banks convex upward. 

Evidently B T (Fig. 20) ® £ . In other words, the tangent at 

P meets the axis of gage heights at a distance 0 T from the 

origin equal to D . The value of D being known, and 

0 T measured on the plot, x can be found, then K, and the 
equation of the corresponding portion of the curve is com¬ 
pletely determined, although the values of x and K may vary 
for different portions. 

Mean Velocity Curve . For each gaging, Q and A becoming 
known, V, the mean velocity past the entire cross-3ection, 
follows from the relation between Q and A, and by plotting a 
series of values of V for different gage heights a curve may 
be determined. As elsewhere stated, this curve is likely to be 
concave toward the axis of gage heights, becoming nearly 
straignt for high stages provided the stream does not over¬ 
flow its banks. 

Let us apply the formula V = C \/R S, and let d represent 

A 

the average water depth in the cross-3ection, = — . Where the 

w 

depth is small compared with the width, as is often the case 
in natural channels, there is no great error in assuming R = d. 

We may then write V = C S^* d^. But d being a function of D, 
we may furtner write as approximately true for those cases in 

which C and S are fairly constant V = C‘ Dir, in which C* is 






















51 


assumed constant. The grapn of 
in Fig. 21. The first deriva¬ 
tive of V witn respect to D 
shows the inclination of tne 
tangont to the curve at any 
point P, witn respect to the 
axis of -gage heights, to be 

C * • It will also be seen 

that the tangent meets the 
axis of velocities at a dis¬ 
tance from the origin equal 

V 

to j , and tne axis of gage 

heights at a distance (-D). 
Under tne assumptions here 
made the curve should be tan¬ 
gent at the origin to tne 
axis of velocities, except 
in case of a bar below the 
gaging station causing dead 
water in low stages, when 
the velocity tends to dis¬ 
appear at some point above 
zero depth. In such case 
the curve is said sometimes 
to show a reversal, and to 
become convex to tne axis 
of gage heights on approach¬ 
ing it. 


such a curve is illustrated 



Discharge 


Curve. Continuing the assumptions already made 


and combining the expressions 
for A and V, we have 

2 jc -ri 

Q = X C 1 D a , the graph 

of which is represented in 
Fig. 22. It will be seen that 
a tangent to this curve at any 
point P should meet tne axis 
of gage heights at a distance 
from the origin equal to 


D. As previously 



noticed, the curve is normally 
convex to tnat axis, and ordi¬ 
narily tangent to it at tne 
origin; but, like the mean ve¬ 
locity curve, it will meet the 
axis at a height above zero in 
the case of dead water being 
caused at the gaging station 
bv a bar. 


Y 


D 



















52 


43. Use of Discharge Curves for Streams of Unstable Bed . 
Attention has already "been called to tne fact (Art. t'nat, 

if bed or banks be unstable, the constancy of tne relation 
ordinarily assumed between gage height and discharge doe3 not 
hold,. Tavernier even says that in many cases it has been 
found for such streams in the Alps that the gage height is 
much more influenced by tne rising or lowering of tne bed than 
by increase or decrease of discharge (see article on Gaging of 
Streams with Unstable Bed; Annales des Ponts et Chaussees, 

1907 - IV). Special caution must therefore be used in apply¬ 
ing discharge curves to such streams. Instrumental gagings 
must be made at relatively short intervals during periods of 
shifting bottom, and some method of interpolation be employed 
for estimating from gage heights the probable values for dis¬ 
charge on intervening days. Two such methods are described 
by Hoyt and Grover (River Discharge, pp. 95 et seq.), and 
are substantially as follows:- 


and 

those re- 
if chronologically consecutive, 


Stout Method . Plotting gage heights as ordinates 
instrument ally-measured discharges as abscissas 
suiting points, especially 
which may be joined by a 
discharge curve of* reason¬ 
able shape, are SO' joined, 
thus establishing an ap¬ 
proximate standard curve, 
such as Curve A of Fig-23, 
in which points given by 
gagings of May 1st, 11th 
and 21st are joined. If 
now on June 5th, at an ob¬ 
served gage height of 13.8, 
an instrumental gaging 
gives Q, = 2300 cubic feet 
per second, we may get that 
value from the standard 
curve by correcting the 
gage by + 1 foot- And if 
on June 20th, at an ob¬ 
served gage height of 11.3, 
a gaging gives Q, = 1150 
cubic feet per second, we 
may get this value from 
the standard curve by cor¬ 
recting the gage height by 
-1.2 foot. By plotting a 
series of such corrections 
as ordinates, and the cor¬ 
responding time intervals 
as abscissas, joining the 
plotted points by a con¬ 
tinuous curve, as shown, 
and assuming the curve to 
show the probable law 



of 


/v$.23. 


change 


cf 


corrections, we 






















53 


may scale cff for any intermediate date the correction to be 
applied to the observed gage height for that day. With the 
corrected gage height we may then read from the standard dis¬ 
charge curve the probable discharge for the day in question. 

Bolster Method . A standard curve is constructed from time 
to time as accumulated gagings seem to justify. If curve A, 

Fig. 23, represent the. standard for present use, and a reliable 
gaging on June 5th give the point shown for that date, then 
the standard curve is shifted bodily downward, parallel to 
its original position, until it pass through the point men¬ 
tioned, as shown by dotted line. In its new position the 
curve is assumed to show the true relation between gage heights 
and discharge for the channel as it then exists. For any date 
intermediate between May 21st and June 5th the curve may be 
moved to the corresponding interpolated position, and the 
probable discharge for the known gage height read directly 
from it in that position. The necessity for this shifting of 
the discharge curve up or down for streams of unstable bed, 
if it is to show correctly at different times the relation 
between gage height and discharge, has been plainl}' noticed 
and called to attention in connection with Mississippi river 
measurements (see Trans. Amer. Soc. Civil Engrs., Vol. 34, 
p. 449). 

44. Methods Available for Measuring Velocity . Although 
many devices are possible, and many have been tried, for meas¬ 
uring velocities in open channels, by far the most of presentr 
day stream gagings are made with the current meter, having a 
wheel which is turned by the force of the current. If prop¬ 
erly rated, and intelligently used, under reasonable condi¬ 
tions, this will give in general greater accuracy than other 
devices, and will give them more quickly and economically. 

Very extensive and valuable series of measurements have 
been made, however, with floats; and for rectangular flumes it 
is probable that no instrument is superior in accuracy to the 
rod or tube float. For sluggish currents, also, in which the 
meter wheel would either not turn at all, or would give uncer¬ 
tain results; and in streams with many floating weeds'or much 
floating debris, which might clog or injure a meter, some form 
of float is in order. 

Modifications of the Pitot tube have been employed some¬ 
what by French engineers. The method has great value for find¬ 
ing the velocity in pipes flowing under pressure, but is of 
limited service in open channels, although it may be useful in 
small, clear streams, especially if the velocity is exception¬ 
ally high. 

The formula V = C \JR S is not infrequently applied to 
stream measurements where direct gagings are impracticable, 
but the results are often subject to large ~/ror. 


















































































































About 1840, large paddle wheels, between 15 and 20 feet 
in diameter , were used for gaging the flow in some of the 
Lowell canals, and are referred to by Francis in "Lowell 
Hydraulic Experiments".' 

Rarely, some kind of coloring matter has been added to a 
small stream, such as that in a sewer, the color permitting 
the eye to note the moment of passage by the upper and lower 
extremities of a section of known length, as in the case of 
floats. 

The plan has been slightly experimented with, also,.of 
mixing a suitable chemical in known volume with a small stream 
and measuring the resulting dilution as an index of the volume 
flowing. 

Observation of the respective temperatures of two small 
streams immediately above their junction, and of the combined 
stream below the junction, has been made the basis of deter¬ 
mining their volumes. 

Sundry other very exceptional means of measurement are 
referred to on p. 17 of Water Supply and Irrigation Paper, 

No. 95, of the U. S. Geological Survey. 

45. Advantages and Disadvantages of Floats . Floats are 
made either single and shallow, to be used at the surface; or 
as a combination of a small surface float and a larger and 
principal sub-surface float, joined by a cord; or in the form 
of a weighted tube or rod, to reach from the stream surface 
as nearly as may be to the bed. For general purposes of meas¬ 
urement they present the following advantages 

(a) Interfere but little with the natural motion of the 
water, even in small streams. 

(b) Measure the velocity directly, and unperceived large 
errors not likely. 

(c) Can be used in streams of any size, and at all veloc¬ 
ities . 

(d) Little affected by silt or weeds. 

(e) Give desired forward velocity, i.e., the component 
of actual velocity at right angles to the section. 

(f) Cheap and easily repaired. 

On the other hand, the following disadvantages are to be 
noticed:- 


(a) Difficult to regulate their course. 











55 


(b) Impelled by adjacent moving mass of water only, ve¬ 
locity of which may be either maximum or minimum of 
the varying impulses due to pulsations. Numerous 
observations therefore necessary for accurate results. 
Cunningham, from his experiments in Ganges canal, 
considered about 50 repetitions of float runs needed. 

(c) If bed is irregular, no form of float can be run 
close to bottom throughout its course. 

(d) At least three men needed for field work, and on 
large streams additional men for boats. 

(e) Several cross-sections may have to be measured to 
determine the mean for the float course. 

(f) Influence of surface float and cord (in case of double 
floats) upon velocity of lower float. In great depths 
exposed surface of cord may exceed that of float; 
thus in some of Humphreys and Abbot’s experiments^ 

on the Mississippi the cord presented an area 50fe 
greater than the lower float. 

It will be seen that in determining vertical 
velocity curves by double floats the influence of 
the surface float tends to carry the observation 
curve wholly outside the true curve if the maximum 
velocity be at the surface; but if the maximum veloc¬ 
ity be below the surface the observation curve tends 
to fall inside the true curve down to the depth where 
the velocity equals the surface velocity, and thence 
downward to fall outside the true curve. 

(g) Uncertainty as to the relative positions of surface 
and sub-surface float. 

(h) Tipping of sub-surface float, changing its exposed 
surface and increasing its liability to be carried 
upward in eddies. 

46. Surface Floats. Surface floats are likely to be used 
but seldom, except in a hasty reconnoissance, or in streams 
swollen by flood and dangerous for more exact methods of gaging. 
They necessarily have some degree of submergence, but this 
should be slight, and there should be as little as possible 
exposure above the water, so as to reduce to a minimum air re¬ 
sistance and the effect of wind currents. In small streams 
it is desirable that the float should itself be relatively 
small. If it be circular it will have the advantage of pre¬ 
senting in all positions the same area to the current. Very 
simple objects may be used; in small streams an orange answers 
well the above requirements, and its color permits the eye easily 
to follow it. In large rivers a flag may be needed to mark the 
float. In streams made unsafe for boats by floating debris or 






























































































































56 


\ 


ice , pieces of drift or cakes of ice may themselves answer 
as a substitute for artificial floats. 

47. Double Floats . Double floats consist of a rela¬ 
tively small surface float and a larger sub-surface float 
joined to the former by a cord and adjusted to run at the 
desired depth. The surface float permits of observing the 
course of the main float and of recovering it at the end of 
the run. Ellis has pointed out the features mentioned below 
as important in a good double float, and they will be seen to 
hold in the form used by him in extensive gagings of the Con¬ 
necticut river at Thompsonville in 1874 and illustrated in 
Pig. 24:- 

(a) The lower float should offer large lateral resist¬ 
ance and this should be equal in all directions, 
on account of rotation. 

(b) It should offer small vertical resistance, so as not 
to be affected by eddies or downward currents. 

(c) It should have sufficient stability of flotation to 
stand upright in the water. 

(d) It should have sufficient preponderance of weight to 
prevent floating upward in eddies and to keep the 
connecting cord vertical, but not such as unduly to 
increase the size of the surface float. 

(e) The surface float should be as light and as small as 
practicable and yet sustain the required weight, and 
its form should be such as to offer minimum resist¬ 
ance to the water and to wind. 

(f) The connecting cord should be the smallest that is 
consistent with requisite strength. 


















Section of Sub'S urface Pioat, 



Top view of S u.b“ Surf ace. Float 



S ide view of Surface. Float 



Top view of Surface Float 


Manner of using Floats. 




































































































































































58 


\ 


• Hpd. or Tube Floats . These consist of wooden rods 
or metal tubes, weighted at the bottom so as to float with 
axis vertical. They should project slightly above the water 
surface, and should extend downward as nearly as practicable 
to the bed, preferably nine-tenths or more of the full water 
depth. For a less relative immersion than nine-tenths the re¬ 
lation is not 30 well known, between the observed velocity of 
the float and the mean velocity of the water for the entire 
depth. Such floats are simple in construct ion, are free from 
the uncertainties attending the use of double floats, give 
directly a close approximation to the mean velocity past a 
vertical, and for channels with smooth bed and not of exces¬ 
sive depth afford a very accurate means of finding the discharge . 

In Mississippi river gagings rod floats as long as 35 or 
40 feet have been used, but greater lengths cannot b8 success¬ 
fully handled. This type is especially well suited to artifi¬ 
cial canals, and for over fifty years tubes have been the 
standard device for gagings in the canals at Lowell. Those 
introduced by Francis were 2 inches in diameter, made of tin 
plates soldered together and we ighted with a solid cylinder of 
lead at the bottom sufficient to sink the tube nearly to the 
desired depth. By dropping in a little additional weight the 
tube could be immersed to the exact depth required, leaving 
about four inches of length above the surface . The top was 
closed with a cork. Latterly brass tubes have been adopted 
for some of the canals . 

Cunningham used 1-inch tin tubes in the Ganges canal 
gagings, while in certain experiments for determining the rate 
of travel of fresh water down the Thames logs about a foot in 
diameter and about 12 feet long, weighted with iron at the end, 
have been employed. 

49. Details of Float Measurements. 


(a) The float should be set free sufficiently far above 
the upper range line to allow it to attain its normal 
position and velocity before reaching the line. 

(b) The length of run from upper to lower range lines 
should be no longer than is necessary properly to 
limit errors of observation. The common range in 
length is from 50 to 200 feet, varying with the 
speed of the current and somewhat with the size of 
the stream, but in a very slow current a much shorter 
distance might suffice . 

(c) The points of crossing the upper and lower range 
lines are commonly fixed by transit intersections 
from the shore in the case of large streams; and 
bv graduated lines or beams spanning the channel in 
the case of small streams or canals. 
















vd) lho timing may be done by a atop watch in the hands 
of an observer, or by a chronometer and chronograph 
witn. electric connections to observers at the upper 
and lower transit lines, or by a similar device with 
electric connections and automatic devices operated 
oy the float itself as it enters upon and leaves its 
measured course (see Report of Chief Engineers, U.S.A 
1883, p. 2226). 

(e) Sub-surface floats are frequently run in series so 
as to obtain in each set velocities in a vertical 
plane at successive intervals of depth, varying from 
one foot, for rivers of moderate depth, to 5 feet 
for deep rivers like the lower Mississippi. 

(f) Pie id notes snould record! names of observers; name 
of stream; locality of gaging, with sketch of site; 
date of gaging; time of beginning and of ending 
gaging; kind of float used; gage height at suitable 
intervals for determining mean value; reference of 
zero of gage to permanent bench mark; test of watch 
for error; remarks as to wind, causes of abnormal 
runs, etc.; and for each run, the depth of immersion 
of float, distance out from reference line to cross¬ 
ing by float of upper and of lower range; transit 
angles, if used for locating float; elapsed time by 
watch in seconds and tenths. Soundings, angles or 
other readings for locating them, and accompanying 
gage heights, also form part of the complete record. 

50. The Current Meter . This instrument has a wheel fur¬ 
nished with vanes or cups, made to revolve by the current, the 
number of revolutions being registered upon a suitable counter 
by means of gearing or by electrical connections, or else in¬ 
dicated by sound. The relation between the number of revolu¬ 
tions per unit of time and the velocity of the current is de¬ 
termined by experimental rating. Because of its convenience 
and the accuracy that may be attained with it under a great 
variety of conditions, the current meter is in general pref¬ 
erable to any other device for measuring velocities in open 
channels. The following advantages are also to be noticed:- 

(a) Observations are confined to one section, and there 
is less need than with floats of a straight reach 
of channel of uniform dimensions. 

(b) Observations can usually be taken more rapidly than 
with floats, a smaller party is required for the 
field work, and computations of discharge are shorter 

(c) Variations of velocity are averaged during the time 
of an observation. 

(d) Observations can be taken tolerably close to the 
bed and banks . 










60 


(©} The instrument admits of the easy and rapid measure¬ 
ment of the mean velocity past a vertical, and of 
the particular velocity at any point. 

(f) At dangerous sites, hy the use of cables and trav¬ 
elers, it is possible for one man to do the entire 

work of gaging from the shore . 

* 

Nevertheless, the current meter is a somewhat delicate 
instrument, and success in its use depends upon the selection 
of a suitable type for the special work in hand, upon its me¬ 
chanical perfection, its condition when used, its proper ma¬ 
nipulation during gaging, and the accuracy of its rating. 
Certain inherent defects, more or less serious according to 
the style of meter and the conditions of its use, as enumerated 
by Ellis, are : - 

(a) Difficulty of rating and liability to change of rate 
if lubrication be varied, or bearings become worn, 
or the instrument be injured. 

(b) Irregular or suspended action at low velocities. 

(c) The meter may register full revolutions only, which 
may be of several feet pitch, in which case the error 
of a partial revolution may be of importance . 

(d) The action of the meter is liable to be impeded by 
floating debris, and to be irregular in water con¬ 
taining sediment . 

(e) If free to take its own direction, the meter is 
likely to stand at a slight angle from the current; 
and also if the current is oblique to the cross- 
section the meter will not give the desired com¬ 
ponent of velocity at right angles to the section. 

(f) Faults of construction, and liability to derangement 
of mechanism. 

In its primary form the current meter was invented about a 
century ago by Woltuiann, a hydraulic engineer of Hamburg. His 
meter had to be removed from the water for each reading of the 
number of revolutions, and it was not till about I860 that 
electrical registration was applied for obviating this. The 
instruments in use at the present time (see Figs . 25-27) vary 
fundamentally in the type of wheel, which in some, such as the 
Haskell, has screw-shaped blades, and in others, such as the 
Price, cups like those of the anemometer. Important types are 
frequently made in different sizes,- small ones, to be rigidly 
attached to jointed metal rods and designed for use in chan¬ 
nels of moderate depth; and larger ones, to be suspended by 
flexible cord or cable and suited to the largest rivers. 

Nearly all forms are arranged for electrical registration; 










































































































































Fig. 2 6. /ias/re// Current Meter 








































































61 


\ 


but some, such as the Eteley, provide also mechanical regis¬ 
tration, through gearing; while one style, the "Price acoustic", 
announces by sound only the completion of each ten revolutions. 

The relative value for general use of the screw or pro¬ 
peller type and of the anemometer type appears not tc be clearly 
settled. Each is capable of doing excellent work. Tn the 
former the wheel axis is horizontal and the bearing surface 
greater than• in the latter, in which the axis is usually ver¬ 
tical, with the wheel revolving in a horizontal plane. The 
friction is relatively greater, therefore, in the propeller 
type than in the anemometer, and the wheel likely to be less 
sensitive to sluggish currents . The Price wheel, which is of 
the anemometer class, is supported upon a small pintle, and the 
construction is such as thoroughly to exclude water and grit 
from the bearing surfaces. The bearing friction is, therefore, 
not only small, but tolerably constant also, even in silt- 
bearing water . 

Both forms show noticeable error ir. registration when 
tipped at an angle to the direction of the current, as is 
likely to happen in working from an unstable support, such as 
a boat; the Price, for example, tending to over-register, and 
the Haskell (which is of the propeller type) to under-register, 
though to a less degree. The propeller wheel of the Haskell 
and the Kajds (Hungarian) types present a somewhat conical 
shape to the current, the resulting tendency of which is to 
throw off leaves and weeds which strike the wheel,- an advan¬ 
tage in streams in which these are running. The anemometer 
style usually has a longer pitch than the propeller, that is 
to say, it turns leas rapidly for a given velocity of current, 
which is advantageous in measuring high velocities, for which 
it is difficult or impossible to count the revolutions mentally 
To the Price meter has even been added a special device by 
which only each fifth revolution is indicated, for use in rapid 
currents . 

The Eteley meter is a small instrument having helicoidal 
vanes and used only upon a rod. It is of fine workmanship, 
sensitive, and has no superior for use in clear water of mod¬ 
erate depth; it is somewhat easily clogged by leaves and grass, 
however, and its bearings do not exclude^ silt or sawdust, so 
that it is not suited to streams carrying much matter in 
suspension. 

The "Colorado" meter, designed especially for use in ir¬ 
rigation ditches, is attached to a rod, has a horizontal wheel, 
with cups, and is arranged tc admit of its approaching very 
close to the stream bed. 

The Price acoustic meter is also to be used with a rod. 

It has a small wheel with six cups, and at every ten revolu¬ 
tions of the shaft a little hammer is tripped and, striking a 
diaphragm, causes a clicking sound, v/hich is transmitted through 








the rod to the observer'e ear. The gurgling of the water, if 
deep, and other noises often obscure the clicks, however, so 
that they cannot be distinguished; otherwise the instrument 
is excellent . 

A meter fixed to a rod is completely under the control of 

the observer as to position in the water, but a suspended meter 

has to be provided with a tail to make it head into the cur¬ 

rent; and to prevent its being carried out of a vertical by 
the current pressure it must be more or less heavily weighted 
'.ordinarily by from 10 to 60 pounds) . The torpedo-shaped lead 
weight now adopted, suspended a little below the meter, is also 
provided with a tail to aid it in heading up stream. Even with 
a heavy weight it is often impracticable to keep the meter in 
the vertical in deep and swift currents, and a guy line from a 
transverse cable a hundred feet, more or less, up stream may 
then be needed. 

In studying tidal currents it is sometimes desired to 
learn not only the speed of the current at chosen depths below 
the surface, but also the direction. For this purpose a direc¬ 
tion attachment is made for the larger Haskell meters, consist¬ 
ing of a closed metal chamber inserted in the line of the meter 
axis, between the wheel and the tail, filled with oil, and in 
which is a compass needle, free to swing into the magnetic 
meridian. There is an electrical connection to a repeater box 
above water containing a compass, over the face of which a 
pointer may be slowly revolved until it has the same direction 
as the meter and current, when it stops and the compass bearing 
may be read. Direction apparatus devised by Leviavsky has 
been applied to studying the direction of sub-surface currents 
in hydraulic experiments in Russian rivers (Eng. News 9 Sept. 1, 
1904) . 

Various methods are in use for ascertaining the number of 
meter revolutions in a velocity observation. That of the 
"acoustic" meter has already been mentioned. When operated 
without an electric current, the Fteley meter registers through 
gearing on a train of dials. Automatic electrical registering 
devices are supplied by the instrument makers, but a much 
simpler and far less expensive arrangement is a small buzzer, 
telephone receiver, or telegraph sounder; it is then necessary 
to count mentally the number of buzzes or taps, but the atten¬ 
tion thus required is a safeguard against otherwise unperceived 
faults in the working of the equipment as a whole. 

The Hydrographic Department of Hungary employs a very 
complete registering apparatus (described in Annales des Pcnts 
et Chaussees, 1698,- III, IV), comprising an endless paper 
band which is automatically unrolled from its spool at a rate 
proportional to the descent of the meter through the water, 
and upon which is indicated, through the action of a specially- 
devised chronograph, each entire revolution of the meter wheel, 
each twentieth, or each hundredth, according to the velocity, 










63 


% 


as well as each half second and each hundredth a second. 

The indication, by sound or otherwise, of the completion 
of successive revolutions of the meter wheel, is effected by 
the- alternate making and breaking of an electric circuit . The 
details vary with the type of instrument, but the principle is 
perhaps sufficiently illustrated in Pig. 28, showing a section 

Section A-A 


a 




of a Price meter. This cut is taken from U. S. G. S. Vater 
Supply and Irrigation Paper, No. 94, in which are also given 
very complete directions for taking apart the meter head and 
replacing worn or injured parts. By means of flexible, insu¬ 
lated copper wires, which may serve at the same time .for sus¬ 
pending a lightly-weighted meter in the water, the instrument 
is put into circuit with the recording device and battery 
above water . A switch is also introduced in the case of an 
automatic register, being thrown in at the beginning of an 
observation, and off at the end. Por a buzzer or telephone 
receiver a small dry cell supplies all the current that is re¬ 
quired. One wire of the circuit is connected to the metal of 
the meter at any convenient point (h in th& figure); the other 
is in this case secured at the binding-pest, d, whence there 
is connection via a delicate platinum spring, a, through the 
center of the hard rubber insulation, c, into the contact 
chamber surrounding the upper part of the meter shaft. This 
part of the shaft has an oval-shaped eccentric, m, which 
touches the spring in revolving and completes the circuit 
onca in each revolution. 
































64 


* 


51* Hating the Meter . A currant meter observation always 
consists in counting, or reading from dials, the number of wheel 
revolutions in a corresponding time interval, which may or may 
not have been decided in advance, the time being ta&en prefer¬ 
ably with a stop watch. The average revolutions per second 
may now be computed, and are then to be translated into velocity 
of the water in feet per second. The relation involves the 
"pitch" of the wheel, or length of stream threads passing it 
for one revolution. With propeller wheels having accurately 
made helicoidal vanes, such as the Fteley, the pitch may be 
found with considerable closeness by measuring appropriate 
dimensions of the wheel, but in general each current meter is 
experimentally "rated" from time to time . 

Almost universally this is done by moving the meter at as 
nearly uniform a rate as practicable over a measured distance, 
say from 50 to 300 feet ordinarily, in still water, and noting 
the number of revolutions and the elapsed time . Runs are made 
at various speeds covering the whole range likely to occur in 
practice, and the results plotted on cross-section paper, with 
velocity in foet per second as one co-ordinate, and revolutions 
per second as the other. A smooth "rating curve" is then passed 
among the plotted points. From this curve a rating table is 
usually drawn off, and is to be adjusted by differences as ex¬ 
plained in Art. 40 for discharge tables. It should give veloc¬ 
ities in general to two decimal places, and low velocities to 
three places. 

Among the more noticeable methods of rating are:- 

(a) Suspending the meter from a light truck moving on a 
track above and either over or at the side of the 
water. Devices may be added for automatically start¬ 
ing and stopping the watch and counter, and for pro¬ 
ducing uniform motion of the truck, although this is 
usually propelled by pushing. 

(b) Suspending the meter from the bow of a boat, and pro¬ 
pelling the boat by motor, cable or oars, over a meas¬ 
ured course, although oars are inferior for this pur¬ 
pose . 

(c) Suspending the meter from the extremity of a horizontal 
arm describing a circle about a pivot, the meter thus 
passing through a circumference of known length. 

(d) . Suspending the meter from a sled and moving it through 

a long slit in the ice on the surface of a pond. 

With the special form of chronograph used by the Hydro- 
graphic Department of Hungary, whereby the relation between 
speed of revolution and speed of advance may be determined for 
successive small distances, even for those corresponding to 
but a single revolution of the meter wheel, it is claimed that 
expensive devices for moving the meter with uniform speed 


* 










65 


* 


thjrough the water are unnecessary, pushing by hand being suf¬ 
ficient; and that by varying the speed at intervals during 
each run & very few trips over the measured course suffice, 
in place of the numerous trips, ranging from 25 to over 100, 
which are usual under the ordinary rating methods. 

In rating, the meter is commonly placed 2 feet or more 
below the water surface (for high velocities it should perhaps 
be still deeper) , and if suspended from a boat must be well 
away from the disturbance caused by the latter,- say at least 
5 feet beyond the bow of an ordinary row-boat . Although with 
a truck an approximate rating of a small meter might be obtained 
in a trough a few square feet in cross-section, for an accurate 
rating a much larger section seems necessary, and a canal lock 
or reservoir is often made use of. 

To guard against error from any small velocity that may 
exist in nominally still water, the runs should b8 made in 
pairs, that is, alternately in each direction at substantially 
the same velocity, and the mean for the pair used in fixing 
the rating curve . 

The rating of a meter may become seriously altered by 
injury to cups or varies, or by change in bearing friction 
through wear or change of contact apparatus, and possibly in 
minor degree by change in lubrication. Experience seems to 
indicate, however, that if the only change be in bearing fric¬ 
tion all plotted ratings of the same meter should give parallel 
curves . A rough method often employed for testing the condi¬ 
tion of the meter is to whirl the wheel in quiet air by a quick 
movement of the hand, and note the time required for coming to 
rest; if the time varies materially from that found immediately 
after rating, a change in rating is presumed. 

A suitable stretch of still water is not always available, 
in which case resort may be had to the inferior method of rating 
in a current, provided it be steady and, preferably, slow. 

Let us suppose, the meter suspended from a boat, and runs made 
in pairs, each pair comprising a run down stream and one up 
stream, both as nearly as practicable at the same absolute 
speed. 

Let V„ and V’ ~ observed absolute velocity of boat and 
a meter in respective runs of any pair, 

V and Vf = their corresponding relative velocity 

with respect to the water, 

V w ~ corresponding absolute velocity of water, 
supposed to be constant . 

Then, with the current, V r = V a - V w , 

and against current, + V w . 

... V r + Vjt = V a + V4, and we may plot for each 


pair the mean value thus found. 


2 









66 


Moving water is sometimes employed also to give a partial 
check upon meter ratings by comparing the observed speed of 
floats with that indicated by meter. A more comprehensive 
check is sometimes practicable where the entire stream dis¬ 
charge may be determined simultaneously by meter and by a 
standard weir . Subject to whatever error may exist in the 
weir measurement, a coefficient may thus bo found for correct¬ 
ing a meter rating made in still water and adapting it to the 
conditions of moving water (see Cornell experiments by E. C. 
Murphy, U .S .G .S . Water Supply and Irrigation Paper, No. 95, 
pp . 80-32) . 

That the conditions are not precisely the same in rating 
a meter in stilL water and in using it to gaga running water 
has been pointed out by Stearns, who calls attention to the 
following differences (Trans. Amer. Soc. Civ. Engrs ., Vol. 12. 
2883) 

,fa) The velocity of the meter through still water may be 
substantially uniform, but the velocity of running 
water in contact with the meter varies through pulsa¬ 
tions, and in integration varies with the changing 
position of the meter. 

(b) In rating, the direction of motion may practically 
coincide with the axis of a screw meter, or the plana 
of revolution of a cup meter; but in a stream the 
water may strike the meter obliquely, because of 
eddies or cross-currents, or faulty holding of the 
meter; and in integration this obliquity is sure to 
occur, since the effective direction of the water 
with respect to the meter is that of the resultant 

of the motion of the current and the reverse of the 
motion of the meter. 

(c) In rating, the forward motion is the same for all 
parts of the meter, while in a running stream the 
water may vary considerably in velocity within the 
diameter of the meter wheel. In Stearns' experiments 
a difference as great as 10 $ existed within a distance 
equal to the diameter of the Pteley meter wheel, near 
the bottom and sides of the Sudbury conduit. 

Assuming now, for simplicity, the case of a propeller 
meter with helicoidal vanes, it will be seen that if there ware 
no solid or fluid friction, and no disturbance of the water by 
the meter frame and vans edges, the wheel , considered as a 
screw, would make the same number of revolutions in traversing 
a given distance through still water, whatever might be its 
forward speed through the water; and the revolutions per second 
would be directly in proportion to the forward speed. A line 
such as a-b (Eig. 29 ) would represent the first-mentioned re¬ 
lation, and o-d (Eig. 30) the second. In fact, however, there 
is a bearing friction such that some noticeable velocity, 


> 

> > 
> > 

> > > 

> 








6 




varying from say 0.5 foot per second down to less than 0.1 foot 
per second with different meters, is required to start the 

wheel revolving. This is represented by the distance o-c . 

* 

The resisting force due to bearing friction does not ma¬ 
terially increase, however, as the velocity through the water 
increases,, but, on the other hand, relatively diminishes as 
compared with the force developed by the latter, the result 
being that its effect in reducing the theoretic number of rev¬ 
olutions rapidly decreases as the speed increases, and at mod¬ 
erate velocities becomes small and approximately constant. 

In consequence the rating line as a whole is concave toward 
the axis of velocities, the curvature being most pronounced 
over the distance c-f, that is, for velocities less than from 
one-half foot to two feet per second, according to the partic¬ 
ular meter or type of meter used. From f on, say to d, or to 
above the velocities usually encountered, the line is either 
practicality straight, or has a very flat curvature. There is 
some resistance to turning due to the fluid friction of the 
water in contact with the whirling wheel, this friction pre¬ 
sumably increasing approximately as the square of the velocity 
and, according to Hajos, becoming sensible above velocities of 
say 10 feet per second, and tending to cause increased con¬ 
cavity, as at d-e . 

A slight change in bearing friction has so large relative 
effect upon revolutions, at very low velocities, and the dif¬ 
ference between discharge as measured by weir and as measured 
by current meter has been found so large at low velocities, 
that it is considered safer not to use the meter for gagings 
in velocities below about one-half foot per second. 



y > > 





























































\ 







































68 


For some meters the rating line as a whole shows so little 
curvature that it is drawn as a straight line among the plotted 
points representing the results of the various observations. 

If the points indicate noticeable departure from a straight 
line, however, it is well to draw the rating line accordingly. 
This line is usually located by eye among the plotted points, 
being drawn by aid of a straight-edge, sprung to a flat curve 
if necessary; but not infrequently the method of least squares 
is applied to finding the equation of the line, assuming the 
typical form y = ax + b for straight lines, and y = a + bx 
+ cx 2 for curved lines, and Murphy considers that this method 
should be used if high accuracy is sought from meter measure¬ 
ments at low velocities. It is perhaps fair to claim that, in 
a first-class rating, plotted points should seldom vary more 
than 1/b from a smooth rating curve . 

It is thought that the results of a rating are best shown 
by a curve constructed as in Fig. 29, and that values can be 
read from it more accurately than from one constructed as in 
Fig. 30; but, for direct use in connection with gagings, either 
a curve of the latter form, based upon revolutions per second, 
or an equivalent rating table, is required. 

52. Retails of Current-Meter Measurements . The observa¬ 
tions may be made either by the method of point measurements, 
as at the assumed depth of mean velocity, or at mid-depth, or 
at a series of points in a vertical; or by the method of inte¬ 
grated measurements, as previously described. Point measure¬ 
ments in a regular current appear to reproduce the conditions 
of rating more nearly than do integrated measurements, and 
therefore have some theoretic advantage. The practical manage¬ 
ment of th'e meter will be mainly governed by the depth and ve¬ 
locity of the current, but the following plans may be noticed:- 

(a) Operating a rod meter by hand while wading, as in 
shallow streams. 

(b) Operating from a bridge, using rod meter or suspended 
meter according to depth of water and height of bridge. 

(c) Operating from a row-boat, either anchored or secured 
to a cross line . The meter may be suspended to advan¬ 
tage from the end of a projecting boom, or lowered and 
raised upon a standing line secured to a heavy weight 
resting on the stream bed. 

(d) Operating from a launch or catamaran, anchored or not 
according to circumstances. This is necessary in 
large and swift rivers. On the Mississippi the meter 
is often suspended by a cable from a boom projecting 
obliquely from the stern of the launch, and is kept 
vertical by being weighted, and by means of a guy line 
running from the meter to the end of a boom in the bow 
of the launch, or to a sheave on a bow anchor and thence 

to the launch. 








69 


(®) A Cal:>lQ is suspended over the stream from bank to bank, 
and the meter is operated from a sling or chair hung 
from^the cable . This method has been used in very 
swixt rivers, where operations would otherwise have 
been dangerous . 

(f) A cable is suspended over the stream, and the meter 
is run out from the shore, being hung from a carrier 
and guyed from a second cable farther up stream. Both 
these methods have been used by the U. S. Geological 
Survey (see 11th Annual Report, 1889-’90, Part 2, p.15). 

The length of un observation should be fixed with reference 
to the time allowable for, and the general accuracy aimed at in, 
the gaging as a whole; and in particular with reference to ob¬ 
taining a mean of the pulsations of the current and to keeping 
relatively small the errors due to vertical motion of the meter 
in integrating, inaccurate timing, or failure to record partial 
revolutions . 

In gagings for the Mississippi River Commission, point 
measurements appear usually to have been continued for from 2 
to 10 minutes, averaging perhaps from 3 to 5 minutes. 

In integration, uniformity of vertical motion is important. 
The observer will be aided in securing this by having an as¬ 
sistant announce regular time intervals corresponding to a 
given rise or descent of the meter, although with practice he 
may secure a sufficiently uniform motion without such aid. 

The rate of vertical motion must also be slow enough to pre¬ 
vent its affecting materially the registration of the meter; 
it must therefore be small as compared with the horizontal ve¬ 
locity of the current, Stearns’ experiments indicating that 
for the Fteley meter 1 to 20 is a safe ratio. 

In integration, the elapsed time must be taken on complet¬ 
ing the vertical, and also the number of revolutions, which for 
nearly all meters can be observed to the nearest integral only; 
while in a point measurement we may attempt to observe either 
the revolutions in an integral number of seconds, or the number 
of seconds for an integral number of revolutions. Since the 
number of revolutions n per second equals the total number N 
during the observation, divided by the number of seconds t, 
it will be seen that when N is large compared with t, as with 
a short-pitch meter in a rapid current, it is the more accurate 
to base the observation upon an integral number of seconds; 
but, when N is small compared with t, to base the observation 
upon an integral number of revolutions. 

There is risk of error if the meter be used near the stream 
surface, experiments by Murphy with the small Price meter, 
largely employed by the U. S. Geological Survey, having shown 
the error to be material if the meter be less than 6 inches 
below the surface in currents of more than 1-J- foot per second 
velocity . 












Accurate results are not to be expected from the meter in 
sluggish currents, and experience has led the U. S. Geological 
Survey not to regard as suitable for a gaging station a cross- 
section showing less than one-half foot per second velocity in 
more than 15^ of the area. 

Field notes should record:- Names of observers; name of 
stream; locality of gaging, with sketch of site; date of gaging 
time of beginning and of ending gaging; name and number of 
meter; gage height at suitable intervals for determining mean 
value and for adjusting soundings; reference of zero of gage 
to permanent bench mark; test of watch for error; distance out 
from reference line to station; water depth; depth of observa¬ 
tion; reading of watch at beginning and end of observation; 
revolutions of meter wheel. If computations are to be made in 
the field, there will also be added, in suitable columns,- 
revolutions per second; velocity in feet per second; width, 
mean depth, and area of vertical strips of cross-sect ion; dis¬ 
charge past strips; and area, mean velocity and discharge for 
entire section. 

53. The Pitot Tube . In its simplest form,- a vertical 
tube, with a right-angled bend at the bottom pointing up stream 
this device was invented by Pitot in 1730. 

The impact of the current against the open¬ 
ing causes water to rise to, and in a steady 
current to remain at, a height above the 
free surface of the stream approximately 
equal to the velocity head at the orifice . 

That is, referring to Fig. 31, 

2 _ 

h = — , nearly,- whence v = \j 2gh . If the 
2g 

elbow be pointed down stream, some suction 
is likely to result and the column within 
the tube to sink even below the free sur¬ 
face outside. 

The value of h gene rail:/ being small, and the free surface 
of a stream more or less disturbed, there are manifestly dif¬ 
ficulties in the way of using this simple form for stream 
gaging . About 1865 Darcy and Bazin de¬ 
veloped the instrument farther by using 
two orifices,- one pointing up stream 
and made very small so as to avoid dis¬ 
turbing the stream lines and to lessen 
the oscillations within the tube, and the 
other, also small, pointing at right 
angles to the current. The principle 
will appear from Fig. 32, where it will 
be seen that h* measures simply pressure 
head, while h" measures pressure head 
plus velocity head, and as before 

h - it . But it is now possible, by /va.32 

2g 









































































































































































































. 





























































71 


applying a moderate auction with the mouth at s, to raise both 
columns to a convenient elevation for reading, without chang¬ 
ing tnoir difference. The perfected instrument has a scale 
for reading the water columns, and 3 top-cocks for holding the 
vacuum and the water columns in taking a reading. 

Tho value of h is so small that but little accuracy is to 
be expected at velocities below about one foot per second from 
vertical water columns. The value of h may be exaggerated, 
however, by inclining the columns, or by using oil in the 
upper part of the tubes in place of air and thus forming a 
differential oil gage (see paper by W. B. Gregory in Trans. 

Amer. Soc. Meehan. Engrs ., Vol. 2b) . The Pitot tube is un¬ 
suited to turbulent water, and has the general disadvantage of 
indicating velocity at a single instant only; but in smooth, 
swift currents, especially in small channels and to depths not 
exceeding say 5 feet, it may often serve a very useful purpose. 

Ritter proposed an ingenious special form for use in meas¬ 
uring surface velocities in flood gagings when working from a 
bridge, the part containing the tips being weighted and pro¬ 
vided with a vane to insure heading into the current, while 
flexible tubes transmitted the pressure to the observer's 
manometer on the bridge . A peculiar feature was the introduc¬ 
tion of coiled capillary tubes between the orifices and the 
flexible transmission tubes, so as to exclude water from the 
latter, compressed air thus becoming the medium for transmit¬ 
ting the pressure to the manometer (Annales des Ponts et 
Chaussees, 1866, 2d semestre , pp. 697 et seq.) . 

Practically, v = c >/ 2gh . For an impact tube alone, if 
given a small, thin-edged orifice, c would doubtless be closely 
unity; but the suction produced by the second, cr pressure tip, 
usually increases the observed h, so that c may fall materially 
below unity. 

Experience indicates that care must be given tc the rating 
cf each instrument of the type of the Pitot tube, especially 
where two orifices and transmission tubes are employed, and 
that the rating should bo performed in conditions ar. nearly 
as possible duplicating those to be met in the actual gaging. 
Thus, experiments by E. C. Murphy (Eng. News, Aug. 12, 1909) 
showed the column difference to vary considerably,for the same 
velocity, as the orifice tip was moved farther and farther from 
the channel bed; and showed the mean velocity in an experimental 
trough, as obtained from a gaging with a tube rated in still 
water, to vary from 3 to 11^» from the value obtained from the 
discharge independently measured. 

Three principal methods have been used in rating such 
instruments:- 

(a) Comparing the readings of the tube with the velocities 
shown by floats run over a definite course . 








72 


(>:) Securing the tube to a boat or truck and drawing it 
through still water at different observed speeds. 

(c) Holding the tij> at many points in the cross-section 
of a channel, the discharge of which could be simul¬ 
taneously measured by other means, such as a weir. 

Darcy used methods (a) and (c), adopting the mean of the 
values thus found. 


54. Measurement of Discharge by Chemical Means . In an 
article by Charles E. Stromeyer in Min. Proc. Instn. Civ. F.ngrs 
1905, Vol. 160, pp. 349 et seq., he describes a chemical method 
of gaging flew which he has used in measuring the supply of 
feed water and circulating water for engines, and with Y/hich 
he has also experimented in small rivers, sev/er outfalls, etc. 
In the experiments for which results are given the error ranged 
variously from between 1 and 2fo to between 7 and 8 fo. Quoting 
from the above article:- ”In gaging by the chemical method, 
a fairly concentrated solution of seme chemical for which very 
sensitive reagents are known is discharged, at a uniform and 


accurately 
analyses are 
the addition of 
chemical in the 


known rate, into the stream under observation, and 
made of the water of the stream before and after 
the chemical . The ratio of the percentage of 
concentrated solution to the percentage added 


to 


the water is obviously the same as the ratio of the volume 
or flew of the water per second to the volume of flow of the 
solution per second. Various salts can be used; in general, 
acids or alkalis would be employed with brackish or sea water, 
and chlorides with hard water 




Let x s- weight of flow of stream per second (not materially 
altered by amount of chemical solution added), 


w = weight cf flow of chemical solution per second, 

r , r , r 3 , = ratios of chemical in approaching stream, 
' " added solution, and departing stream, re¬ 

spective ly, 


Then x (r 3 - r / ) = r 2 
. * x a r z W c 

r 3 “ r / 

The conditions necessary to success by this method are that 
the flow be sufficiently swift and agitated to ensure thorough 
mixing of the chemical with the stream in a moderate distance, 
and that between the two sampling stations there be no marked 
variation in impurities other than that due to the chemical 
purposely Introduced. Colorimetric analysis has been suggested 
for use,*by which a suitable coloring matter, eosine for ex¬ 
ample, should be added, and the degrees of dilution determined 
by comoarison of samples with standards of known dilution, but 
experiments by Belcher and. Colscn, M.T .T ., 1907, indicated that 























73 


reasonable accuracy in estimating the dilution could be obtained 
only by ad’&ing an excessive amount of coloring solution; they 
concluded that for streams of any considerable volume gravi¬ 
metric analysis would be necessary for accurate results, and 
that more chemical skill would be required than is ordinarily 
possessed by the engineer . 

In exceptional cases a suitable coloring solution has 
been injected into the stream flowing in a closed conduit, 
serving as a substitute for floats, the color enabling the eye 
to note the time of arrival at some manhole or other opening, 
and thus affording means of approximately determining the mean 
velocity for the whole cross-section. 

55. Measurement of Relative Discharge by Use of Thermometer . 

For the quick approximate measurement of relative" discharge , 
especially of small mountain streams whose turbulence would 
forbid the use of meters or floats and for which the establish¬ 
ment of weirs is for any reason inadvisable, the thermometer 
may sometimes be employed. The method was described at length 
by C , Fitter in a paper entitled "Emploi du Thermometre, dans 
la Jaugeage des Petits Cours d'Eau" (Annales des Ponts et 
Chaussees, 1884, pp. 323 et seq.). 

It is to be applied at the junction of two streams whose 
temperatures are sensibly different, but uniform throughout 
the croes-section of each, as well as throughout the cross- 
section of the main stream below' the junction . Three stations 
are therefore made use of,- one on each separate stream shortly 
above their junction, and one on the combined stream far enough 
below the junction to permit thorough mixing to have teen ef¬ 
fected. Bitter employed thermometers graduated to fifths of 
a degree Centigrade, and took the temperature of the water 
which he removed from the stream in a small wooden pail having 
no metal trimmings . He did not consider the operation good 
unless several successive attempts gave exactly the same results. 

Let Q. = volume per second flowing in one stream above junction, 
q » = » it w ,f o the r " ” 

t and f = their respective temperatures, 

T * common temperature below junction. 

Then, bv the law' of mixtures, OL. = - • 

Q» T - t* 

If the absolute volume at any one of the three stations 
be known from other data, then not only the relative but also 
the absolute values of Q and Q* become known . Fitter cites two 
cases in which volumes of about 4 and about 40 cubic f©et per 
second, respectively, were measured by this means with errors 
of about 2/o and 6^, respectively. 











74 


56 • Computation of Discharge by Means of Slope Formulae . 

Prom the general formula V = C \/r S it appears that, if C , R 
and S can he accurately learned, V and hence Q, can he deter¬ 
mined without a direct instrumental gaging. For this purpose 
a length 1 must he chosen, great enough if practicable to keep 
within satisfactory limits the relative error in determining 8, 
and in which the slope and cross-secticn are tolerably uniform; 
the fall h in water surface for this distance 1 (which, strictly- 
speaking, is to he measured along the slope) is to he found 
from simultaneous readings on gages of known elevation placed 

at the extremities of the reach, whence S = ^ . The average 

hydraulic mean depth R for the reach 1 must also he found, and 
that proper value assigned to C. 

The practical difficulties of measuring S, which is usually 
small, with sufficient accuracy and refinement, and of choosing 
the correct value for C„ are so great that the resulting value 
of Q is often liable to large error, and the method does not 
commend itself except when direct gagings are out of the ques¬ 
tion. It is sometimes applicable, however, as a last resort 
in determining what was the approximate discharge of a river 
in a flood that has passed, marks along the hanks giving the 
means of roughly ascertaining the slope. 

5-7. Accuracy of stream Gagings . An absolute test of the 
accuracy of a stream gaging must generally involve a simulta¬ 
neous independent measurement of the volume flowing, made by 
some method known to he of superior accuracy. This is seldom 
practicable on a large scale, and one must resort to such evi¬ 
dence as is afforded by comparison of gagings made at the same 
time with different meters, or with meters and floats. Many 
such comparisons have been instituted, to a few of which ref¬ 
erence will he made . 

In 1374, extensive gagings of the Connecticut river at 
Thompsonville, Conn., were made by Gen. Theodors G. Ellis, 
using both double floats and current meters. The observations 
were scattered over a period of about four months, during which 
the discharge ranged, in round numbers, from 4,000 to 63,000 
cubic feet per second. One of the most important conclusions 
reached from a study of the gagings was stated to be, "That 
both the floats and meters give concordant results, showing 
that each is reliable; when carefully and intelligently employed, 
and that each method has certain advantages over the other 
under peculiar circumstances” (Annual Report, Chief of Engrs., 
U.S.A., 1878, App. 3, 14, p. 11). 

In the discussion of William Starling’s paper on "The 
Discharge of the Mississippi River”, Arthur Hider cites ten 
gagings made in pairs on the lower Mississippi in 1392 t each 
pair comprising a current meter gaging made in the forenoon 










75 


and Redouble-float gaging made in the afternoon of the same 
day, the discharge ranging for the entire period, in round 
numbers, from 1,000,000 to 1,500,000 cubic feet per second. 

Both sets of observations in each pair were taken at 6/10 
depth. Usually, though not always, the meter gave a less dis¬ 
charge than the floats, the differences ranging as a whole from 
about 4/b downward nearly to zero (Trans. Amer. Soc . Civil 
Engrs., Vol. 35, p. 330). 

As the result of experiments made by James B. Francis 
to test the degree of uniformity attainable in gagings with 
tube floats in rectangular canals, he concluded that, "¥e must 
infer from these seven experiments that any single measurement 
is liable to be erroneous to the amount of 1 fo, or perhaps rather 
more; and in any two experiments the errors may be in opposite 
directions, in which case they may vary from each other 2/C, or 
rather more (Lowell Hydraulic Experiments, p. 200) . 

The writer’s experience with students making for the first 
time tube float measurements in rectangular canals in which 
the discharge varies say from 700 to 1500 cubic feet per second, 
simultaneous gagings being made by independent parties, is that 
the difference^ in discharge for the two gagings of a pair sel¬ 
dom exceeds 2/C , and averages not more than 1-^?. 

Very elaborate experiments were made in 1900 by E. C. 
Murphy, of the U. S. Geological Survey, in the Cornell Univer¬ 
sity flUme, for the purpose of testing the accuracy to be ob¬ 
tained by current meters used under different methods. The 
discharge ranged, roughly, from 200 to 235 cubic feet per sec¬ 
ond. In no case out of 50 comparative measurements did a meter 
discharge differ from a corresponding weir discharge by as much 
as 5nor did two simultaneous meter discharges vary from each 
other by as much as 5 fo. The tests were thought to show that, 
under favorable conditions, discharge can be measured with a 
Small Price mater (one of the types used in the experiments) 
with an error of not more than 1 ;%, although in ordinary river 
sections so high a degree of accuracy could not be expected 
(Trans. Amer. Soc. Civil Engrs., Vol- 47, pp. 370 at seq.). 

E. E. Haskell, an authority upon stream gagings, in dis¬ 
cussing the paper above referred to, expressed the opinion 
that ’’The Niagara river, and similar streams of comparatively 
permanent regimen, can be measured, with residuals not exceed¬ 
ing 2% of their flow”; and that "The discharge of the Lower 
Mississippi River, with its ever-changing conditions, can be 
measured, with residuals not exceeding 5% of its flow.” 

It must not be forgotten, however, that if such accuracy 
is to be obtained with the current meter, then, as stated by 
Haskell, the instrument "must be of good design, well made, 
well rated, well cared for, and used by a careful, painstaking 
observer 


































































58. Tine Required for Gagings . So long as the rate of 
flow of a stream remains constant, no limit need he set to the 
duration of a gaging, except that resulting from considera¬ 
tions of economy. But, in general, streams fluctuate so rap¬ 
idly in discharge that it is desirable to finish observatio.ns 
for velocity in as short a time as allowable, in order to 
avoid error from a change occurring in velocities throughout 
the cross-section during the progress of the measurement. 

The time allowed, therefore, for a current-meter gaging, even 
of a large river, such as the Niagara or lower Mississippi, 
seldom exceeds from 2 to 4 hours. A very complete gaging with 
rod or tube floats can be made in an hour or less in a canal 
of 50 feet width, and a meter measurement by the method of 
vertical integration can be accomplished in even less time; 
a point measurement is likely to require more time than one by 
integration, but dependent upon the number of observations in 
each vertical. 

59. Gagings in Ice-Covered Channels . It is often re¬ 
quired to measure tne discharge ot rivers and canals when 
frozen over during the winter season. The ice covering pro¬ 
duces a large surface friction, which greatly alters the dis¬ 
tribution of velocities from that existing during open-water 
conditions, thus forbidding the use of the ordinary reduction 
coefficients. When the covering is continuous and smooth, 
fairly const suit, velocity relations are found to hold, upon 
which gagings may safely be based, and station rating curves 
may also properly be developed. But if a stream be covered 
with broken and tilted ice, or contain much needle ice, tne 
velocity relations become too variable and uncertain for use. 

Because of the greater 
friction and the resulting 
smaller velocity, tne carrying 
capacity of a given wetted 
cross-3ection is lessened by 
surface freezing, and the dis¬ 
charge can be maintained only 
by an increase of section and 
consequently of gage height. 

After a tnick ice sheet has 
been formed, however, it may 
sufficiently withstand pres¬ 
sure from beneath to permit 
of tne stream flowing with 
moderate pressure head, and 
even with greater velocity 
than prevailed in the same 
cross -3ecti on when uncovered. 

A marked effect of the ice 
covering is seen in the ver¬ 
tical velocity curves, the 
surface velocity being so re¬ 
duced and the concavity so 
increased that two points of 
mean velocity are found in 
each vertical (Fig* 33)• 
















77 


Gagings of ice-covered streams can be made most accurately 
by observing velocities at sufficient points in each vertical 
to define the vertical velocity curves. Vertical integration 
may often be employed to advantage, although the formation of 
an ice coating on meter cable or rod is troublesome at low 
temperatures. Where enough vertical velocity curves have been 
determined to establish reliable reduction coefficients, sub¬ 
sequent observations may safely be made, on streams having a 
smooth ice cover, at one or two points only in each vertical, 
as in the case of open streams. 

» 

A study of about 350 vertical velocity curves, taken in 
various ice-covered streams and in different conditions, has 
shown the average position of the two threads of mean velocity 
to be very closely l/lO and 7/l0 depth, respectively, measured 
from bottom of ice ('J.S.G.S. Water Supply and Irrigation 
Paper, No. 187, p. 79). There appears to be greater constancy, 
however, in velocity relations in the central part of the ver¬ 
ticals. The reduction coefficient to be applied to the velocity 
at mid-depth, reckoned from bottom of ice, to give mean velocity 
past the entire vertical, averaged 0.88 for tne curves above 
referred to, and varied but little at any single gaging station 
with change of stage. For two observations in a vertical, 

2/l0 and 8/l0 depth below bottom of ice have been found to 
give excellent results, the mean of tne velocities at those 
depths giving, on tne average, very exactly tne mean velocity 
past the entire vertical. 

Comparatively few stat.ion-rating discharge curves for 
conditions of ice covering have thus far been constructed. 

They must evidently lie, on tne whole, nearer the axis of 
gage heights than the corresponding curves for open water 





























































78 


# 


conditions, and will assume different positions accordingly 
as gage heignts are measured to tne lower surface of tne ice, 
or to tne water surface in a hole cut in the ice (see Fig. 34, 
from U.S.G.S. Water Supply and Irrigation Paper, No. 187, 
p. 43). The effect upon such curves of varying thickness of 
ice, or varying roughness of surface, has not been well de¬ 
termined. 





















?COPY. DEL. TO CAT. DIV. 







NOV 6.11909 


































































































































